AP Physics 1 Curriculum Module Rotational Motion PROFESSIONAL DEVELOPMENT AP ® Physics 1 Rotational Motion CURRICULUM MODULE For the redesigned course launching fall 2014 The College Board New York, N[.]
Rotational Inertia
• How is rotational inertia related to the inertia discussed in the context of Newton’s first law of motion?
• How is the rotational inertia of an object or system related to the structure of that object or system?
• How does the rotational inertia of an object or system affect the object’s or system’s motion?
This lesson introduces rotational inertia, helping students understand how it influences rotational kinetic energy and angular momentum Building on their prior knowledge of translational inertia related to mass, students learn that rotational inertia describes an object's resistance to changes in its rotational motion For example, objects with larger rotational inertia require more torque to initiate movement and also need greater torque, like frictional torque, to bring them to a stop.
Friction forces significantly influence both translational and rotational motion by exerting opposing forces and torques that slow down or stop moving objects For instance, as a wheel rotates, frictional forces generate torques that eventually cause the wheel to halt, illustrating the parallel between translational friction and rotational frictional torques Emphasizing these similarities helps students connect familiar concepts of linear motion with the new, often unfamiliar, principles of rotational motion, fostering a deeper understanding of the physics involved.
X Connections to the Curriculum Framework
Learning objectives related to the topic of rotational inertia covered in this lesson are identified below:
The learning objective (4.D.1.2) focuses on enabling students to develop effective data collection strategies to accurately predict torque, angular velocity, angular acceleration, and angular momentum Students will learn to consider whether these variables are treated as clockwise or counterclockwise rotations, ensuring precise measurements and analysis Mastery of these strategies is essential for understanding rotational dynamics and applying physics principles reliably.
I = kmr 2 with respect to a well-defined axis of rotation, and refine the research question based on the examination of data [See Science Practices 3.2, 4.1, 4.2, 5.1, and 5.3]
This learning objective focuses on two key aspects: first, describing a model of a rotational system by understanding its rotational inertia and internal structure, and second, applying this model to analyze how angular momentum changes during interactions with other objects or systems In this lesson, the emphasis is on understanding and describing rotational systems through their rotational inertia, which reflects the system’s internal structure The second part of the objective will be explored in Lesson 3, where rotational inertia is used to analyze angular momentum changes resulting from interactions between rotating objects or systems Incorporating these concepts enhances our understanding of rotational dynamics and the principles governing angular momentum.
The learning objective (5.E.2.1) focuses on students’ ability to describe or calculate the angular momentum and rotational inertia of a system based on the positions and velocities of its constituent objects Students are encouraged to develop qualitative reasoning skills with complex, compound objects while performing quantitative calculations involving both extended objects and point masses These competencies align with Science Practice 2.2, emphasizing a comprehensive understanding of rotational dynamics in physical systems.
This learning objective applies to two lessons in this module: the description of rotational inertia and angular momentum It clarifies the scope of systems involved, such as compound objects like a hammer, which has a specific rotational inertia Students may be asked to analyze how shortening the hammer's handle affects its rotational inertia, recognizing that it decreases as the distance \( r \) in the inertia equation becomes smaller Additionally, scenarios like a wheel with the axis through the center might be used to explore how replacing a rubber rim with a lead rim increases the rotational inertia due to the increased mass \( m \) These examples help students understand the factors influencing rotational inertia in different systems.
As a result of this lesson, students should be able to:
• Explain how mass, radius, and internal structure can be used to describe the rotational inertia of an object or system
• Relate how rotational inertia affects the motion of an object or system
Students should have a solid understanding of basic concepts of objects in translational motion, including key principles of movement and forces Additionally, they need to be familiar with concepts related to torque and rotational motion, as outlined in the curriculum framework, to grasp the mechanics of objects in rotation effectively.
• Fundamentals of torque and equilibrium
• Distinctions between translational and rotational equilibrium
• Kinematics with rotational quantities and their symbols (t, a, w, q)
• Changes in rotational acceleration and rotational velocity due to torques exerted on the object or system
• Conversions between linear and angular quantities
• Experimental design and data collection/analysis related to torques
You can use Handout 1, “Using the Symbols of Rotational Motion,” as a formative assessment to check which of these ideas should be reviewed with students prior to this lesson
X Common Student Misconceptions and Challenges
A common misconception is that rotational inertia is solely a property of an object’s mass, similar to linear inertia However, rotational inertia depends not only on an object's mass but also on how that mass is distributed around the axis of rotation Therefore, objects with identical masses, such as petri dishes, rotating batons, and eggs, can exhibit different rotational behaviors Emphasizing this concept is crucial in Lesson 1 activities to help students understand the importance of mass distribution in rotational motion.
Furthermore, it’s important to show students that changing the axis of rotation
Understanding that grasping a baton at its end, rather than the middle, alters its rotational inertia is crucial; rotational inertia depends on how mass is distributed relative to the axis of rotation Students often mistakenly believe that an object exhibits rotational inertia only when rotating, but it is important to emphasize that rotational inertia is a property with respect to a given axis and remains consistent whether the object is stationary or in motion Recognizing this helps clarify fundamental concepts in rotational dynamics and improves comprehension of how mass distribution influences an object's rotational behavior.
• Petri dishes with lids or similar closed, flat cylinders (four or multiples of four)
• Steel balls (four per petri dish or cylinder, with the diameter of each ball equal to the height of the container)
• Two pieces of ″ schedule 40 PVC pipe (each cut to a length of 1.5 m)
• Four end caps to fit the PVC pipes
• Four “concrete anchors” (can be found in hardware stores; should fit firmly inside the PVC pipes without sliding)
• Several eggs, some raw and some hard-boiled
• Commercially available solid disk-and-hoop set
• Varied objects to determine rotational inertia (e.g., basketball, baseball, wooden dowels of varied diameters and lengths)
• Handouts 1 and 2 and Appendix C (Note: Materials for handouts and appendices are listed separately on those documents.)
Lesson 1: Ro ta tional Iner tia
Activity 1: Mass Distribution and Rotational Inertia, Part 1
This activity provides students with an informal understanding of how mass distribution influences rotational inertia To prepare, create sets of four petri dishes or similar cylindrical containers, painting their interiors opaque to conceal the contents Inside each dish, glue four identical steel balls in different arrangements—such as spaced evenly around the perimeter or aligned across the diameter—to illustrate various mass distributions Seal the dishes securely to prevent students from seeing inside, allowing them to explore how different mass placements affect rotational behavior.
Figure 1: Suggested setup for mass distribution and rotational inertia, part 1
Students first confirm that all dishes have the same mass before working in teams to develop experimental methods for qualitatively analyzing their internal structures They observe how each dish behaves differently when rolled on their sides, with variations in behavior linked to differences in rotational inertia For larger classes, creating multiple identical sets allows students to compare results effectively, aiming to identify each dish's unique configuration This activity enhances understanding of rotational dynamics and moment of inertia, making it an engaging experiment for physics education.
Activity 2: Qualitative Lab — Introduction to Rotational Inertia
The "Qualitative Lab — Introduction to Rotational Inertia" series of investigations is designed to strengthen students' understanding of rotational inertia These experiments encourage students to make initial predictions, describe and analyze their observations, and compare their results with their original hypotheses By engaging in this process, students deepen their comprehension of rotational inertia and develop essential scientific reasoning skills.
Students have the opportunity to write their responses to lab questions and submit them for review, fostering individual understanding Additionally, small-group representatives can share their responses with the entire class, serving as a foundation for engaging class discussions This approach encourages active participation and enhances collaborative learning experiences.
Lesson 1: Rotational Inertia is a fundamental concept in Physics 1, highlighting its importance in understanding rotational motion When evaluating student responses, it is crucial to identify misconceptions related to rotational inertia and address them effectively Providing targeted explanations and opportunities for repeated activities enable students to correct misunderstandings and reinforce their comprehension of the concept Addressing common misconceptions among students helps improve learning outcomes and ensures a clearer grasp of rotational inertia.
Handout 3 (“Formative Assessment on Torque and Rotational Kinematics”) also contains questions that may be used for formative assessment
Activity 3: Demonstration — Mass Distribution and Rotational Inertia, Part 2
This demonstration features two identical batons, made of the same materials and with equal mass, but differing in their rotational properties due to their distinct construction It illustrates key concepts of rotational inertia and how a baton’s design influences its spin and stability By comparing their behavior, viewers can better understand the physics of rotational motion and the impact of shape and distribution of mass This simple experiment effectively demonstrates the principles of rotational dynamics, making complex concepts accessible and engaging.
Handout 2, in which students use two metersticks, one with identical masses attached to the ends and one with identical masses attached closer to the middle
Rotational Kinetic Energy
• How is total energy of a rotating object calculated?
• How does torque do work to change the energy of a rotating system?
In this lesson, students learn how to describe and calculate the total energy of a rolling object by combining translational and rotational kinetic energies, such as a ball rolling across the floor They also explore how torque does work to change an object's rotational kinetic energy, applying the work-energy theorem to analyze energy transfer in rotational systems.
X Connections to the Curriculum Framework
In this lesson, students will explore the concept of rotational kinetic energy to deepen their understanding of how energy is distributed within rotating objects and systems By achieving these learning objectives, learners will be able to explain how rotational kinetic energy contributes to the total energy of an object, enhancing their comprehension of energy conservation and dynamics This knowledge is essential for a comprehensive analysis of energy transfer in physical systems, making it a key concept in physics education.
• Learning Objective (5.B.4.2): The student is able to calculate changes in kinetic energy and potential energy of a system, using information from representations of that system [See Science Practices 1.4 and 2.1]
Understanding the interaction between a system and its environment is essential, as the environment can exert a force on the system, performing work that alters the system’s energy Specifically, when a force from the environment acts on a system, it can change the system's total energy, which includes both kinetic and potential energy components Students should be able to make informed claims about these interactions, demonstrating comprehension of how external forces influence energy transfer within physical systems, in accordance with Science Practices 6.4 and 7.2.
As a result of this lesson, students should be able to:
• Explain how external forces exerted on a system can exert torques to change the state of rotation of the system
• Design an experiment to explore the effect of external torques on rotational kinetic energy
To analyze the energy changes within a system, it is essential to consider rotational kinetic energy, translational kinetic energy, and potential energy Calculating these requires knowledge of the system’s angular or linear velocity, which directly influences the energy state The rotational kinetic energy can be determined using the formula \( \frac{1}{2} I \omega^2 \), where \( I \) is the rotational inertia and \( \omega \) is the angular velocity Similarly, translational kinetic energy is calculated as \( \frac{1}{2} m v^2 \), with \( m \) representing mass and \( v \) the linear velocity Additionally, potential energy depends on the system's height within a gravitational field To accurately evaluate the changes in energy, it is crucial to use the appropriate formula for rotational inertia, which accounts for the distribution of mass in the system Understanding these energy principles and applying the formulas allows for precise computation of energy variations during rotational and linear motion.
This lesson assumes that students have the following prerequisite skills and knowledge from the previous lesson or from earlier work in the course:
• Familiarity with concepts and equations for rotational kinematics,
Handout 1 calculate rotational kinetic energy equation tables in Appendix B.)
(See the AP Physics 1 particularly rotational velocity (w) and rotational inertia (I), in order to
Understanding how to relate linear velocity (v) to angular velocity (ω) is essential for converting between linear and rotational motion for a point on a rotating object This relationship allows for accurate calculations of linear speed based on the angular velocity, excluding the center-of-mass speed during combined rotational and translational motion Mastering these conversions is key in various physics and engineering applications involving rotational dynamics.
• Experience with the work–energy theorem as it relates to linear motion, in order to form the analogous relationship for angular motion
Use Handout 1, “Using the Symbols of Rotational Motion,” as an individual assessment to verify students’ understanding of the prerequisite concepts If students lack this knowledge, the handout can be repurposed as a collaborative small-group activity to ensure all students grasp the foundational principles of rotational motion.
X Common Student Misconceptions and Challenges
A common misconception is that an object with rotational motion cannot also have translational motion, or vice versa Additionally, many overlook the crucial role of friction in generating rotational movement For example, if a ball is placed on a frictionless ramp, it will slide down without rolling, highlighting that friction provides the torque necessary for the ball to roll instead of merely sliding.
• Appendix E and Handout 3 (Note: Materials for handouts and appendices are listed separately on those documents.)
Activity 1: Demonstration — Rotational Kinetic Energy
A rotating object or system has rotational kinetic energy that is calculated in a method analogous to translational kinetic energy In translational motion,
Lesson 2: Rotational Kinetic Energy kinetic energy depends on translational inertia or mass (m) and linear speed (v):
2 In rotational motion, kinetic energy depends on rotational inertia (I)
Of course, an object can be moving in linear and rotational speed
(or translational) motion and angular (or rotational) motion at the same time, so the total kinetic energy is the sum
To illustrate the concept of energy transfer, imagine rolling a basketball across the floor and asking students to describe its energy As the ball rolls to a stop, the gravitational potential energy remains unchanged, while the total kinetic energy consists of both rotational and translational components When the ball eventually stops, the mechanical energy is transformed into thermal energy through friction, as the molecules of the ball and floor generate heat due to negative work done by the frictional force.
Activity 2: Lab — Using Rotational Kinetic
Energy for the Ball on a Ramp Lab
Students can revisit previous experiments involving rolling objects by designing a lab that integrates both linear and angular motions, emphasizing rotational kinetic energy This approach helps students better understand the interplay between translational and rotational dynamics, enhancing their grasp of rotational motion concepts Refer to Appendix E, “Lab — Using Rotational Kinetic Energy for the Ball on a Ramp,” for detailed instructions on incorporating rotational energy analysis into your lab activities, improving overall comprehension of physics principles.
In this experiment, students observe a ball rolling down a ramp and off a tabletop to analyze energy conversion, specifically focusing on the fraction of initial potential energy that transforms into rotational kinetic energy versus translational kinetic energy They design their own experiments using available materials, making decisions on measurement techniques, relevant calculations, and the optimal number of trials to ensure accurate results This hands-on approach helps students understand the principles of energy conservation and rotational dynamics through practical experimentation.
Laboratory work serves as an effective form of formative assessment, allowing students to apply their knowledge through designing and conducting related investigations These hands-on activities help evaluate students' understanding and readiness to progress to the next lesson Incorporating laboratory experiments as follow-up assessments enhances student engagement and provides valuable insights into their practical skills and conceptual comprehension.
To assess students’ understanding of rotational kinetic energy, utilize the prompts and questions in Appendix E to develop a comprehensive set of assessment questions If many students struggle with these questions, provide additional explanations of the concept to reinforce their understanding Follow up with targeted activities on rotational kinetic energy, review students’ responses, and only proceed to the next lesson once they demonstrate mastery.
Lesson 2: Ro ta tional Kine tic ener gy
Students are expected to submit comprehensive lab reports that include detailed analysis such as calculations, observations, and conclusions Instructors will evaluate these reports and provide individualized feedback—either directly on the report or via a rubric—focusing on proper terminology and accurate calculations Feedback will also identify any misconceptions, guiding students to revisit the lab for further observation or to revise their conclusions if misconceptions may hinder their progress in subsequent lessons.
Post-lab peer reports, where each lab group presents their conclusions orally or on a whiteboard, facilitate valuable feedback on experimental design and results, promoting a constructive peer-review process This activity encourages students to reassess their conclusions, make necessary amendments, and explore alternative approaches, ultimately enhancing their understanding Allocating an additional 20 to 30 minutes for this activity allows sufficient time for meaningful discussion, depending on class size and the number of reporting groups.
Changes in Angular Momentum and Conservation
Lesson 3: Changes in Angular Momentum and Conservation of
• How are rotational collisions analogous to collisions of objects or systems in linear motion?
• How does torque change angular momentum?
• How do objects or systems interact to change angular momentum?
This lesson explores the fundamental concepts of linear momentum and its conservation, demonstrating their similarities to angular momentum and the conservation of angular momentum Students will understand how linear momentum behaves in collisions and interactions, highlighting the importance of conservation laws in physics By drawing parallels between linear and angular quantities, the lesson enhances comprehension of momentum principles, essential for mastering classical mechanics.
X Connections to the Curriculum Framework
This lesson focuses on key learning objectives including making predictions, providing detailed descriptions, and designing experiments effectively Emphasis is placed on developing strong writing skills, whether in paragraph format or within laboratory reports and journals By the end of Lesson 3, students will be able to formulate accurate predictions, craft clear descriptions, and design well-structured experiments, all while honing their scientific writing abilities.
Students will learn to predict the behavior of rotational collision scenarios by applying analysis methods similar to those used for linear collisions This involves understanding the analogy between impulse and the change in linear momentum, as well as between angular impulse and the change in angular momentum Mastering these concepts enables accurate prediction of rotational collision outcomes, aligning with science practices 6.4 and 7.2 for comprehensive physics education.
Students will learn to identify and justify the appropriate mathematical methods for calculating changes in angular momentum caused by torques, even in unfamiliar contexts or when using diverse representations beyond equations, ensuring a solid understanding of rotational dynamics.
Students will develop the ability to design effective data collection and analysis strategies to examine the relationship between torque and changes in angular momentum, aligning with science practices such as data collection, analysis, and interpretation This learning objective emphasizes understanding the fundamental physics of rotational motion and ensuring students can apply experimental methods to investigate torque's effect on an object's angular momentum, fostering skills in scientific inquiry and critical analysis.
Learn to describe and utilize various representations to analyze how multiple forces acting on a rotating system of rigid objects influence its angular velocity and angular momentum This skill aligns with Science Practices 1.2 and 1.4, enabling students to understand the complex interactions affecting rotational motion in physics.
Students will learn to develop effective data collection strategies to accurately predict torque, angular velocity, angular acceleration, and angular momentum by considering the direction of rotation—clockwise or counterclockwise—relative to a clear axis of rotation They will also refine their research questions through careful analysis of collected data, ensuring precise understanding of rotational dynamics.
The learning objective (4.D.2.1) focuses on enabling students to describe a model of a rotational system and apply it to analyze scenarios involving changes in angular momentum resulting from interactions with other objects or systems Mastering this concept allows students to understand how rotational forces and interactions influence angular motion, aligning with key science practices such as modeling and analysis (Science Practices 1.2 and 1.4) This skill is essential for comprehending rotational dynamics and explaining real-world physical phenomena involving angular momentum transfer.
The learning objective (4.D.2.2) focuses on enabling students to develop a comprehensive data collection and analysis strategy to assess changes in the angular momentum of a system Students will learn how to relate these changes to interactions with other objects and systems, fostering a deeper understanding of rotational dynamics This aligns with Science Practice 4.2, emphasizing the importance of designing experiments and analyzing data to interpret physical interactions effectively By mastering this skill, students can better evaluate the principles governing angular momentum and the effects of external influences on rotational motion.
Students will be able to apply appropriate mathematical routines to calculate initial or final angular momentum, the change in angular momentum of a system, or the average torque and duration over which torque is applied, essential for analyzing situations involving torque and angular momentum Understanding these calculations is crucial for mastering the concepts in rotational dynamics and meets the criteria outlined in Science Practice 2.2 for effective scientific reasoning.
Learning Objective 4.D.3.2 focuses on enabling students to design an effective data collection strategy to investigate the relationship between a system’s change in angular momentum and the product of average torque and the duration of torque application This goal emphasizes the importance of planning experiments that accurately measure how applied torque influences angular momentum over time, aligned with Science Practices 4.1 and 4.2 to ensure systematic and precise data gathering.
Students will understand how to qualitatively predict the angular momentum of a system in scenarios with no net external torque, aligning with Learning Objective 5.E.1.1 This skill involves analyzing how angular momentum changes or remains constant in isolated systems, enhancing their grasp of conservation principles in rotational motion Mastering this concept supports a deeper understanding of physics science practices, specifically 6.4 and 7.2, which focus on applying reasoning and evidence-based predictions in physical contexts.
Students will learn to accurately calculate the angular momentum of a system when the net external torque is zero, enhancing their understanding of rotational dynamics This skill aligns with Learning Objective 5.E.1.2 and emphasizes the importance of applying conservation principles in physics Mastering these calculations supports students in comprehending how angular momentum remains constant in isolated systems, which is fundamental for mastering rotational motion concepts This knowledge is essential for understanding real-world applications involving systems under zero external torque, fostering a deeper grasp of physics principles.
In this learning objective (5.E.2.1), students will be able to describe or calculate the angular momentum and rotational inertia of a system based on the positions and velocities of its constituent objects They are expected to perform qualitative reasoning with complex, compound objects and to carry out calculations involving both fixed extended objects and point masses This competency aligns with Science Practice 2.2, emphasizing analytical skills in rotational motion analysis Understanding these concepts is essential for mastering angular momentum and rotational inertia in physics, enabling students to analyze the rotational behavior of various systems effectively.
Lesson 3: Changes in Angular Momentum and Conservation of Angular Momentum
As a result of this lesson, students should be able to:
• Calculate net torque on a system and use torque to calculate change in angular momentum
• Plan experiments or data collection strategies or analyze data for changes in angular momentum due to torque exerted on the system
• Apply the concept of conservation of angular momentum to interactions of objects
In readiness for this lesson, the student should be able to:
• Use kinematic equations for angular motion
• Describe or calculate rotational inertia for an object or extended system
• Design experiments and analyze data related to rotational kinematics
• Make analogies between linear and angular motion
• Use torques to determine changes in rotational motion
Use Handout 1, “Using the Symbols of Rotational Motion,” as an individual assessment to verify students' understanding of key prerequisite concepts If needed, this handout can serve as a small-group activity to ensure all students acquire the necessary foundational knowledge Lessons 1 and 2 are designed to cover the essential prerequisites, and teachers can identify any gaps through students' lab reports and class discussions that may require additional review.
X Common Student Misconceptions and Challenges
Using the Symbols of Rotational Motion
Using the Symbols of Rotational Motion
Complete the following table Some of the cells are already filled in for you
Linear or Translational Angular or Rotational
Concept Symbol Formula Unit Symbol Formula Unit
Rotational Inertia for Common Objects, I = kmr 2
Hoop (central axis) or solid object moving in a circle I = mr 2
Hoop (rotating around its diameter) I =1
Solid cylinder or disk (central axis) I = 1 2 mr 2
Linear or Translational Angular or Rotational
Concept Symbol Formula Unit Symbol Formula Unit
Use two identical rods (or metersticks) and attach two masses to each of them as shown (For the masses, clamps or small weights attached with tape work well.)
On each rod, the two masses should both be placed the same distance from the center But on Rod 2, both
Qualitative Lab — Introduction to Rotational Inertia
Qualitative Lab — Introduction to Rotational Inertia
Objects resist changes in motion, and objects with greater mass exhibit higher inertia, making it harder to start or stop their movement For example, on a low-friction surface, it requires more effort to initiate movement of a heavy brick compared to a small wooden block Once in motion, it is also more difficult to stop the brick than the lighter block, highlighting that a brick’s higher inertia resists changes to its existing motion.
Objects in rotational motion, such as a bowling ball, exhibit greater resistance to changes in their spin compared to lighter objects like a basketball It is more challenging to initiate spinning a bowling ball in place and equally difficult to stop its rotation, illustrating that the bowling ball has higher rotational inertia This concept highlights how rotational inertia influences the ease of starting and stopping an object's spin, with heavier objects resisting changes in their rotational motion more than lighter ones.
This lab explores rotational inertia through a series of investigative experiments designed to enhance understanding of this fundamental physics concept Participants will follow structured procedures, making predictions beforehand and analyzing their observations afterward to comprehend why the mathematical equations for rotational inertia are valid The laboratory exercises emphasize critical thinking by encouraging students to reflect on the results and deepen their conceptual grasp of rotational inertia, a key property influencing rotational motion in physics.
Materials: Two identical rods or metersticks, clamps or small weights and tape, broom
To conduct the experiment, you will need a rod or meterstick Start by holding the rod near one end and using your wrist to spin it back and forth in a horizontal plane, swinging it through approximately 20 degrees before reversing direction Next, repeat the process while holding the rod near its middle Your teacher will demonstrate these steps, and to minimize the effect of gravity, you can support the rod with the lab bench during the swings.
If you let your wrist exert equal effort both times, then you apply the same torque both times
Make a prediction: In which case will the rod spin back and forth more easily? Why?
Do it: What did you observe during this investigation?
Follow-up discussion: Why was the rod easier to spin back and forth in one case compared with the other?
In our experiment, mass handouts should be positioned farther from the center compared to Rod 1 to observe their effects on rotational motion Hold each rod at its center and spin it back and forth through a small angle, similar to the procedure illustrated in section A, to analyze how the placement of masses influences rotational dynamics Properly positioning the masses and following the specified spinning technique are essential for accurate results in studying rotational inertia and balance.
Make a prediction: Which rod will be harder to spin? Why?
Do it: What did you observe during this investigation?
Follow-up discussion: Can you explain all the results you’ve seen so far?
C A friend says: “Rotational inertia is just a fancy name for an object’s mass when the object is rotating
If the object has more mass, it has more rotational inertia, and if it has less mass, it has less rotational inertia — end of story Nothing else matters.”
In what ways, if any, do you agree with your friend? In what ways, if any, do you disagree?
In this activity, you'll spin a light broom back and forth in a horizontal plane, similar to spinning a rod You will perform the spin first while holding the broom at the brush end and then at the handle end This exercise helps demonstrate rotational motion and the influence of mass distribution on rotational inertia.
Make a prediction: In which case will the broom be easier to spin back and forth? Why?
Do it: What did you observe during this investigation?
Follow-up discussion: Explain what you observed, and relate your explanation to your explanations from part 1
The diagram illustrates a broom with points labeled B, C, and D, where point C is equidistant from both ends of the broom In this activity, you will hold the broom at one of these points and spin it horizontally, as practiced previously Allowing an end of the broom to slide across the lab bench or floor helps reduce imbalance issues when the broom is out of balance in your hand This experiment demonstrates the effects of mass distribution and balance on rotational motion.
Make a prediction: At which point should you hold the broom so that it will be easiest to spin it back and forth? Why?
Do it: After testing your prediction, describe what you observed How can you be sure that your prediction about what would happen didn’t influence your observations?
Follow-up discussion: Explain your observations in a way that fits in with your earlier explanations in this lab
Rotating the broom around a vertical axis passing through its center of mass is easier than rotating it around a horizontal axis because the distribution of mass and the resulting moment of inertia differ between these axes When turned about a vertical axis, the mass is concentrated closer to the rotation axis, leading to a smaller moment of inertia and requiring less effort to spin In contrast, rotating around a horizontal axis involves the mass being distributed farther from the axis, increasing the moment of inertia and making the rotation more difficult This difference reflects how an object's mass distribution significantly influences rotational ease, as covered earlier in this lab.
By examining part 2B and holding the broom at each of the three labeled points, you’ll notice that the point around which the broom rotates most easily also corresponds to the point where the broom feels most balanced in your hand This demonstrates the relationship between balance and ease of rotation, highlighting the importance of the broom's center of mass for optimal handling and stability Understanding this concept can improve your grip and control when using a broom for cleaning or other tasks.
This article analyzes the rotational behavior of two identical sticks, each composed of half wood and half steel, with different pivot points—Stick A pivoted at its wood end and Stick B at its steel end Both sticks are subjected to identical torques, prompting a comparison of their angular accelerations The key question is, which stick, if any, experiences a greater angular acceleration under these conditions? Understanding how the position of the pivot point influences rotational motion is essential for applying principles of torque and moment of inertia in rotational dynamics.
A The rod should be easier to spin when held near its middle because the rotational inertia of the rod is smaller with respect to an axis through its center
B Rod 2 has a higher rotational inertia because its mass is distributed farther from the axis of rotation, especially when considering the clamps This increased rotational inertia makes it more difficult to spin compared to other objects As discussed earlier, objects with larger rotational inertia require more effort to accelerate angularly, highlighting why Rod 2 is harder to spin.
A more massive object generally has greater rotational inertia, as highlighted by the friend’s observation However, Sections A and B reveal that two objects with identical mass can possess different rotational inertias This variation occurs because rotational inertia depends on how mass is distributed relative to the axis of rotation, with greater inertia resulting when a larger portion of the mass is located farther from the axis.
For optimal spinning performance, the broom should be harder to spin when held by the handle end, as this increases the rotational resistance This is because, in this position, a larger percentage of the broom's mass—the brush—is farther from the axis of rotation, enhancing the moment of inertia and making spinning more difficult Proper understanding of mass distribution and leverage can improve your control and efficiency when using a broom.
The broom is easiest to rotate when held at point D, as this point serves as the axis of rotation Holding the broom at D positions most of its mass, primarily located in the brush, closer to the axis This reduces rotational resistance, making the broom easier to turn and maneuver efficiently Properly balancing the broom at point D enhances ease of use and minimizes effort during operation.
C The broom is easier to rotate around the vertical axis because, in that case, all its mass is within 15 or 20 cm of the axis of rotation
The balancing point, known as the center of mass, is the position where the average distance of the broom's mass bits is minimized, ensuring optimal balance When more mass is concentrated closer to the axis of rotation at this point, the rotational inertia decreases, making it easier to spin the broom around the center of mass Consequently, the broom is easier to rotate around the center of mass (point D) compared to points B or C, due to the reduced rotational inertia in that position.
Formative Assessment on Torque and Rotational
Formative Assessment on Torque and Rotational Kinematics
A string is attached to a nearly frictionless wheel with a diameter of 1.0 meter, and a 20 N force is applied at a 60° angle to the tangent The torque exerted on the wheel by the string depends on the component of the applied force perpendicular to the radius, which can be calculated considering the angle and force magnitude By analyzing the force application at 60° to the tangent, the effective force contributing to torque is determined, allowing for accurate calculation of the torque exerted on the wheel.
2 A basketball with a mass of 0.60 kg and radius of 7 cm is rolling across a level floor at a constant speed of 2.0 m/s
(a) Determine the ball’s angular velocity
(b) What is the ball’s angular acceleration?
(c) How many turns will the ball make in 2 seconds?
A bicycle wheel with a radius of 0.5 meters and a mass of 3.0 kg was rotating at 20 rpm when the rider applied the brakes The wheel continued to turn for an additional 10 revolutions before coming to a complete stop This scenario highlights the importance of understanding rotational motion and deceleration in bike safety and brake performance.
(a) What is the wheel’s angular acceleration?
(b) How far has the wheel traveled across the surface?
A uniform meterstick balances horizontally on a pivot at the 35 cm mark, with an 800-gram object hung at the 15 cm mark and a 350-gram object at the 70 cm mark By applying the principles of torque and equilibrium, we can determine the meterstick's mass The calculation involves setting the clockwise and counterclockwise torques equal to each other, considering the positions and masses of the objects and the meterstick's center of mass Solving these equations reveals the total mass of the meterstick, providing insight into the distribution of weight and balance in the system.
A uniform wooden beam with a mass of 20 kg extends horizontally from a wall, supported by a cable that connects from the far end of the beam to the wall at a 40° angle At the end of the beam, a sign weighing 5 kg is hanging, creating additional load on the structure Understanding the forces and torques involved is essential for analyzing the stability of the beam and ensuring safe support.
(a) Find the tension in the cable that helps to support the beam and sign
(b) Find the horizontal and vertical components of the force the wall exerts on the wooden beam
(b) zero (The ball is moving at constant speed.)
3 First, find the initial angular speed in radians per second:
Then substitute into the equations and solve for angular acceleration and linear displacement:
5 (a) Let L equal the length of the beam:
(Note: Use the component of the tension perpendicular to the beam at the point of attachment
The length of the beam is not needed because L cancels in every term.)
(b) The horizontal and vertical components of forces exerted on the beam must balance
Formative Assessment on Rotational Inertia, Kinematics, Kinetic Energy, and Momentum
Formative Assessment on Rotational Inertia, Kinematics, Kinetic Energy, and Momentum
A light string connected to a 4m mass passes over a frictionless pulley and wraps around a vertical pole of radius r, as depicted in Experiment A When the system is released from rest, the descending block causes the string to unwind and the pole to rotate The apparatus includes a horizontal rod of length 2L with small masses of mass m attached at each end, and its rotational inertia is assumed to be 2mL².
(a) If the downward acceleration of the large block is measured to be a, determine the tension T in the string, in terms of the acceleration of the falling block
(b) Determine the torque exerted on the rotating pole by the string, in terms of the mass of the blocks and the acceleration
When the large block has descended a distance D, the instantaneous rotational kinetic energy of the apparatus can be compared to the value 4mgD to analyze its dynamic behavior Typically, the rotational kinetic energy at this point depends on the angular velocity gained during the descent, which can be related to the potential energy lost To determine whether it matches or differs from 4mgD, one must evaluate the energy distribution between translational and rotational forms, considering the moment of inertia and the torque involved This comparison provides insight into the efficiency of energy transfer in rotational systems and helps understand the underlying physics of the apparatus during its descent.
Greater than 4mgD Equal to 4mgD Less than 4mgD Now consider the experiment again, this time including the rotational inertia of the small pulley
(d) If the rotational inertia of the pulley were large enough to have an effect on the experiment, would you predict your answers to be different? Explain your response
(e) How will the angular velocity of the rotating apparatus and linear velocity of the falling mass compare now with their values calculated in parts (a), (b), and (c)?
(f) Discuss how the torque exerted by the string on the rotating apparatus changes the angular momentum of the apparatus ΣF = ma mgD K 11 (4m v)
(a) First express Newton’s law for the motion of the large block as it falls:
Then substitute the values for the forces on the block, making the downward motion positive: 4mg − =T 4ma
(b) Calculate the torque the tension force exerts on the rotating pole: τ = Tr = (4mg − 4ma ) r
The decrease in gravitational potential energy of the large block is directly converted into both translational and rotational kinetic energy Specifically, the reduction in potential energy equals the sum of the large block's translational kinetic energy and the rotational kinetic energy of the apparatus This energy conservation principle highlights the transformation of gravitational potential energy into kinetic forms during the system's motion.
The rotational kinetic energy of the apparatus is less than 4mgD because part of the gravitational potential energy is converted into the translational kinetic energy of the falling large block Since some energy is used to rotate the pulley, the change in rotational kinetic energy decreases, leading to lower linear and angular accelerations Consequently, all answers differ because the energy distribution affects the motion dynamics of the system.
In part (a), the tension varies along the string, with greater tension on the falling block compared to the rotating apparatus due to the net tension force required to rotate the small pulley In part (b), although the mass (m), gravity (g), and radius (r) remain constant, the acceleration (a) decreases because the falling mass must now accelerate both the rotating apparatus and pulley In part (c), while the gravitational potential energy lost by the mass falling a distance D remains unchanged, some of this energy is converted into rotational kinetic energy of the pulley, indicating energy redistribution during the process.
(e) Because the acceleration is less, as described in part (d), if the same falling distance D is used, then the final velocity is also less:
If the final linear velocity of the falling mass is less, then the final angular velocity of the rotating device is also less: v = ω r
(f) Just as constant net force exerted on a system changes its linear momentum at a steady rate
, the constant torque exerted by the string on the apparatus increases its angular momentum at a steady rate:
Summative Assessment
The horizontal uniform rod, measuring 0.60 meters in length and weighing 2.0 kg, is hinged at the left end to a vertical support, allowing it to swing freely The right end of the rod is supported by a tensioned cord that forms a 30° angle with the rod, with the tension measured by a lightweight spring scale Additionally, a 0.50 kg block is attached to the right end of the rod, influencing its equilibrium and dynamics.
(a) On the diagram below, draw and label vectors to represent all the forces acting on the rod Show each force as a vector originating at its point of application
(b) Calculate the reading on the spring scale, which represents the tension in the cord
When the cord supporting the rod is cut near the end, and the block is removed before releasing the rod to rotate about the hinge, the initial linear acceleration of the rod’s center of mass depends on the forces acting on it Since the rod is released without the tension from the cord, the gravitational force primarily influences its motion The rotational inertia of the rod about the hinge plays a key role in this dynamic Under these conditions, the initial acceleration of the rod's center of mass is less than g, because part of the gravitational force contributes to rotational acceleration rather than linear acceleration, leading to an initial linear acceleration below the acceleration due to gravity.
When a hinged rod is released from a horizontal position without any cord or block attached, the angular acceleration of its end is greater than that of its center of mass because the end experiences a larger torque relative to the pivot point Similarly, the linear acceleration of the rod's end is greater than that of its center of mass due to the higher angular acceleration at the end, resulting in a greater tangential acceleration.
In this laboratory experiment, a solid disk with an unknown mass and a known radius R serves as a pulley, with a small block of mass m attached via a string wrapped around it multiple times When the block is released from rest, it falls a distance D in a time t, providing a basis to analyze the system's dynamics The analysis involves expressing all relevant quantities—such as acceleration, tension, and the pulley’s moment of inertia—in terms of R, m, t, D, and fundamental constants, to derive formulas for the acceleration of the block and the angular velocity of the pulley during the fall.
(a) Calculate the linear acceleration a of the falling block in terms of the given quantities
To determine the acceleration of the block, plot the measured time (t) against the height (D) on a graph, with the height on the x-axis and time on the y-axis, then draw the best-fit line through the data points Label the axes clearly to represent the variables accurately Using the graph, analyze the slope of the best-fit line to calculate the acceleration, as the slope corresponds to the rate of change of time with respect to height This method allows for a precise determination of the block's acceleration based on the recorded data.
(c) The value of acceleration found in part (b)iii, along with numerical values for the given quantities, is used to determine the rotational inertia of the pulley:
The discrepancy between the measured rotational inertia of the pulley and the calculated value can be attributed to additional factors such as bearing friction, slight misalignment, or the presence of residual mass from the support structure These factors increase the overall rotational inertia beyond the theoretical calculation based solely on the pulley’s mass distribution Understanding such differences is crucial for accurate data analysis and precise mechanical measurements in physics experiments.
Turntable suspended on a center post
To determine the rotational inertia of the lightweight, low-friction cylindrical turntable, a reliable experimental method involves measuring its angular acceleration under known torque conditions First, attach a low-mass pulley and a known mass to the turntable's edge, allowing the mass to fall vertically and exert a torque Using a motion sensor or tachometer, record the angular acceleration of the turntable as the mass descends Applying Newton's second law for rotation, \( I = \tau / \alpha \), where \( I \) is the rotational inertia, \( \tau \) is the torque (calculated from the known mass and pulley radius), and \( \alpha \) is the measured angular acceleration, researchers can accurately compute the turntable's rotational inertia Ensuring the setup minimizes friction and accounts for any uneven density distribution enhances the precision of the results Diagrams illustrating the setup with the mass, pulley, and turntable, combined with a detailed explanation of the data collection process, support the experimental procedure for accurate determination of the rotational inertia.
To conduct this experiment, I will utilize essential equipment such as a digital multimeter, ruler, and data logger The digital multimeter will be used to measure voltage, current, and resistance accurately, following standard procedures—voltage and current measurements will be taken using the multimeter’s probes connected in series or parallel as appropriate The ruler will measure physical dimensions like the length and width of the sample, ensuring precise data collection The data logger will record measurements over time to analyze variations and trends The experimental setup will include a power supply connected to the sample, with the multimeter configured for the specific measurement, and diagrams will illustrate how each component is interconnected to ensure clarity This detailed description ensures that another student can replicate the experiment accurately.
To determine the rotational inertia, I will first record the angular acceleration of the rotating object using precise timing methods and angular velocity measurements By measuring the applied torque, either through known applied forces or motor specifications, I can utilize the rotational form of Newton’s second law (τ = Iα) to calculate the moment of inertia I will plot graphs of torque versus angular acceleration to verify the linear relationship predicted by theory, ensuring accuracy in the calculation Detailed steps include measuring the torque, obtaining angular acceleration from sensor data, and then solving for rotational inertia (I = τ/α) These measurements and plots will provide clear, reproducible data that enable other students to perform similar calculations and produce comparable graphical representations, reinforcing the understanding of rotational dynamics.
When assessing the uncertainty in rotational inertia, the measurement of the object's mass distribution or radius likely contributes most to the overall uncertainty, as small errors in these measurements significantly impact the calculated inertia To minimize this uncertainty during experimental procedures, it is essential to use precise measuring tools such as calipers or laser measurement devices for radius measurements and ensure rigorous calibration of mass measurements Improving measurement accuracy in these key areas reduces the overall uncertainty in rotational inertia calculations, leading to more reliable and precise experimental results.
1 2(2 kg)(9.8 m/s 2 ) + (0.5 kg)(9.8 m/s 2 ) = T(sin 30°) t = Ia, τ Mg( )1 2 L 3g α = = I 1 3 ML 2 2L a = aR = a(21 L) a = (2L)(3g)/2L 4g
(b) Show that all net torques and forces must equal zero for equilibrium ΣF x = 0 ΣF y = 0 Στ = 0
Set the pivot at the left end of the rod and solve, using the torque equation
(c) With the attached cord and block removed, the only torque to accelerate the rod is the weight of the rod Since calculate angular acceleration first
Since the answer requires linear acceleration,
The acceleration is less than g
In rotational motion, the angular acceleration is uniform for all points on a rod since it moves as a single object sweeping the same angle in each time interval Meanwhile, the linear acceleration of a point on the rod varies depending on its distance from the hinge, with points farther from the pivot experiencing greater acceleration Specifically, because the end of the rod is twice as far from the hinge as its center of mass, its linear acceleration is twice that of the center of mass, highlighting the relationship between distance from the pivot and linear acceleration in rotational dynamics.
2 (a) Start with the general equation for the motion and then substitute D for and 0 for
To analyze the equation effectively, recognize that when D is treated as a function of t², the equation exhibits a linear form, meaning that plotting “D versus t²” will yield a straight line Students should utilize the provided data table to plot D values on the y-axis and t² values on the x-axis, then construct a best-fit line through the data points to determine the slope Alternatively, plotting “t² as a function of D” is also acceptable, with the slope in this case corresponding to 2/a, facilitating the calculation of the related parameter.
Data points on the graph are given below:
2.0 1.99 iii Since the equation is of the form the graph of “D as a function of t 2 ” has a slope equal to
Students can graph D as a function of t to determine the best-fit line, where the slope represents the value of a, providing an accurate method to find the parameter Although this approach yields correct results, it can be more challenging for students to interpret the relationship between D and t compared to other methods Using a graph to analyze D versus t enhances understanding of the underlying relationship and supports accurate parameter estimation in data analysis.
(c) Examine the equation for rotational inertia to determine what experimental results in measurements of m, R, or a would cause the value of I to be calculated lower than the actual value:
Students are not responsible for knowing or deriving this equation It is important to understand that the ratio g/a is greater than 1 because the tension in the falling block causes it to accelerate at a rate less than g Small values of mass (m) or pulley radius (R), or a large acceleration (a), can lead to underestimated calculations of rotational inertia from experimental data Potential sources of error include wrapping string around the pulley, which increases the effective radius and results in larger torque and thus an underestimated inertia, and slipping of the string, which causes higher acceleration Friction at the pulley does not cause an increase in acceleration; instead, it would decrease acceleration.
3 (a) The response to part (a) should include the following:
To accurately measure angular momentum, apply a controlled set of torques to the turntable using a practical method such as attaching a hanging mass to a string wrapped around the turntable's center post or outer edge Allow the mass to fall from rest and measure the time it takes to fall a specific distance, enabling the calculation of the torque exerted on the turntable This approach provides reliable data for analyzing rotational motion and understanding angular momentum dynamics.