Hafez a radi, john o rasmussen auth principles of physics for scientists and engineers 10

216 7 Linear Momentum, Collisions, and Center of Mass24 A ball of mass m1= 0.5 kg, moving along the x-axis with a speed of 5 m/s, has an elastic head-on collision with a target ball of m

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216 7 Linear Momentum, Collisions, and Center of Mass

(24) A ball of mass m1= 0.5 kg, moving along the x-axis with a speed of 5 m/s, has

an elastic head-on collision with a target ball of mass m2= 1 kg initially at rest What is the velocity of each ball after the collision?

(25) A ball of mass m1and velocityv1undergoes an elastic head-on collision with

a second ball of mass m2 initially at rest Then m1 rebounds with a speed

v

1= −0.5v1 Find the value of m2in terms of m1.

(26) A croquet ball of mass m1= 1 kg and velocity v1undergoes an elastic head-on

collision with a second ball of mass m2that is initially at rest Then m1moves with a velocityv1 and m2moves with a velocityv2 = (4/5)v1 (a) Find the value of m2 (b) Find the relation between v

1andv1 (c) What fraction of the

original kinetic energy goes to the second ball?

(27) Find the fraction of kinetic energy lost by a neutron of mass m1= 1.01u when

it undergoes an elastic head-on collision with a stationary nucleus of: (a) a hydrogen atom(1

1H) of mass m2= 1.01 u, (b) a heavy hydrogen atom (2

1H) of mass m2 = 2.01 u, (c) a carbon (12

6 C) atom of mass m2 = 12.00 u, and (d) a

lead atom(208

82 Pb) of mass m2= 208 u.

(28) A block of mass m1= 1 kg slides along a frictionless horizontal surface with

a speedv1= 4 m/s toward a stationary second block of mass m2= 0.5 kg The

second block is connected to an elastic spring that is not stretched and has a

spring constant kH= 100 N/m The other end of that spring is fixed to a wall,

see Fig.7.35 (a) Is the collision elastic? Explain your answer (b) What will be the maximum compression of the spring?

Fig 7.35 See Exercise (28)

(29) A block of mass m1= 2.5 kg slides along a frictionless horizontal surface

with a speedv1= 8 m/s toward a stationary second block of mass m2= 7.5 kg.

A massless spring with spring constant k = 1,920 N/m is attached to the near

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side of m2, as shown in Fig.7.36 (a) Is the collision elastic? Explain (b) What

is the speed of the mass-spring system at the maximum compression? (c) What will be the maximum compression of the spring? (d) What will be the final velocities of the two blocks?

Before collision

1

After collision

2

m

2

m

1

m

1

m

At maximum

compression

1

2=0

mass-spring system

1

(30) Repeat Exercise (29), this time with m1= 7.5 kg and m2= 2.5 kg.

(31) Show that the fraction of kinetic energy transferred to the target in Example 7.5 is independent of the value of the speed of the projectile,v1 Then, redo this example when m2= 3m1.

(32) A hockey puck traveling at 30 m/s on a smooth ice surface is deflected by

θ1= 30◦from its original direction when it collides elastically with a second stationary identical puck The second puck acquires a velocity atθ2= 60◦from the original velocity of the first puck, see Fig.7.37 Find the speed of the pucks after the collision

Before

collision

x y

At rest 1

Hockey pucks

x

y

1

θ

2 θ

2= 0

′

1

′

2

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218 7 Linear Momentum, Collisions, and Center of Mass (33) If each angle in Exercise (32) is equal to 45◦, then show that only the application

of conservation of momentum is enough to find the speed of the pucks after the collision Also, show that any other two equal angles (say 30◦) are physically

unacceptable

(34) A ball of momentum→p

1collides with an identical stationary ball The first ball deflects by an angleθ1from its original direction with a momentum→p

1while the second ball deflects by an angleθ2from the original direction of the first ball with a momentum→p

2 If the two balls have an elastic collision, then use

the conservation of momentum vector diagram of Fig.7.38and conservation

of kinetic energy to show that the two balls will always move off perpendicular

to each other, i.e.θ1+ θ2= 90◦.

Before

After

x y

At rest

x

y

1 θ

1 θ 2

θ

1 θ 2 θ 0

=

m

1

p

′

1

p

′

1

p

′

2

p

′

2

p

1

p

2

p

Subsection 7.3.2 Inelastic Collisions

(35) In a ballistic experiment like the one shown in Example 7.6, a bullet causes

the pendulum to rise to a maximum height h1= 1.3 cm A second bullet of the same mass causes the pendulum to rise to a maximum height h2= 5.2 cm.

Express the speed of the second bullet as a multiple of the first bullet (36) Find a formula that gives the fractional change of kinetic energy,(K f −K i )/K i ,

in terms of m and M in the first stage of the ballistic pendulum of Example 7.6 Calculate this fraction for m = 10 g and M = 0.5 kg.

(37) A 15-kg mass is moving along the positive x-axis at 30 m/s ,and a 5-kg mass is

moving along the negative x-axis at 50 m/s The two masses collide head-on and

stick together (a) Find their velocity after the collision (b) Find the fractional change of kinetic energy

(38) A car of mass m1= 1,000 kg moving with a speed v1 collides with another

stationary car of mass m2= 2,200 kg The two cars stick together after the

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collision Both drivers had their brakes locked throughout the incident, see Fig.7.39 The police officer measures the skidding distance d to be 2 25 m and

estimated the coefficient of kinetic friction between the tires and the road to

be 0.8 (a) What was the speed of the oncoming car? (b) Find the fractional

change of kinetic energy lost during the impact period

Before collision Just after collision

v

At rest

1

d

(39) A nucleus at rest spontaneously disintegrates into two nuclei, one of which has double the mass of the other Assume that the total mass is conserved before and after disintegration (a) Find the relation between the speeds of the two fragments (b) If 6× 10−17J of energy is released in this disintegration, how much kinetic energy does each nucleus acquire?

(40) Two objects having the same mass m= 5 kg collide Their velocities before collision are→v1= (3→i +6→j )(m/s) and→v2= (−2→i +1→j )(m/s) After collision,

the first object acquires a velocity→v1 = (−1→i + 4→j )(m/s) (a) What is the

final velocity of the second object? (b) How much kinetic energy is lost or gained in this collision?

(41) A stationary radioactive nucleus decays into three fragments Two of these fragments are emitted perpendicularly to each other and have momenta

|→p

1| = 5 × 10−23kg.m/s and |→p

2| = 1.2 × 10−22kg.m/s Find the

mag-nitude and direction of the third fragment

(42) A ball of mass m1= 2.4 kg moving horizontally with a speed of v1= 3 m/s

collides (not head-on) with a second ball of m2= 1.5 kg moving in the opposite

direction with a speedv2= 5 m/s, see Fig.7.40 The first ball bounces off the second ball with an angleθ1= 60◦ and speed v1= 1.5 m/s (a) What is the

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220 7 Linear Momentum, Collisions, and Center of Mass final velocity of the second ball? (b) How much kinetic energy is lost in this collision?

1 2

1 '

2

' θ

60°

Before collision After collision

1

m

2

m

2

m

1

m

(43) Two identical putty balls move along a frictionless floor, as shown in Fig.7.41 Their velocity vectors→v1 and→v2 make an angleθ and move with the same

speed, i.e.v1 = v2 The two balls stick together after collision (a) Use the

momentum vector diagram shown in the figure to prove that the magnitudev

and the directionφ of their common velocity→v are given by:

v =1

2v1

√

2+ 2 cos θ, φ = sin−1sinθ/√2+ 2 cos θ

(b) Takingv1= v2= 20 m/s and θ = 45◦to calculatev, φ, and the fractional

change in the kinetic energy

1

Before

x y

θ

2

m

After

x

y

m

φ

m

y

φ θ

θ

1

p

2

p

Section 7.4 Center of Mass (CM)

(44) An oxygen atom(16

8 O) has a mass mO= 16 u, and a carbon atom (12

6 C) has a mass m = 12 u The average distance between their nuclei in the CO molecule

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is d = 0.113 nm, see Fig.7.42 How far from the oxygen nucleus is the center

of mass of the molecule?

x

y

CM

(45) For the system of particles shown in Fig.7.43 and when d = 1 m, find the location of the x and y components of the center of mass Does your answer depend on the value of m? Explain.

x

y

4 m

x= −d x=0 x= +d

y=d

(46) Three particles, each of mass m, are located at the corners of an equilateral triangle of side a, as shown in the Fig.7.44 Show that the center of mass of the system lies on a common point on the three lines that connect each vertex with the midpoint of the opposite side (the medians)

x

y

CM

0

a a

a

(x a= / 2,y a= / 2 )

m

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222 7 Linear Momentum, Collisions, and Center of Mass (47) In an ammonia molecule (NH3), the three hydrogen (1

1H) atoms are at the corners of an equilateral triangle of side a = 0.16 nm that forms the base of a

pyramid, with nitrogen atom(14

6 N) at the apex above the center of this triangle

by h = 0.037 nm, see Fig.7.45 Find the distance of the center of mass of the ammonia molecule above the plane of the hydrogen atoms

H

a

h

(48) A mass m1= 2 kg is connected to a mass m2= 3 kg by a massless rod The

location of m1and m2are given by the position vectors→r

1= (4→i + 5→j )(m)

and→r

2= (2→i + 3→j )(m), respectively Find the coordinates of the center of

mass

(49) Three uniform thin rods, each of length L, are arranged to form the shape shown

in Fig.7.46 The vertical arms have mass M and the horizontal arm has a mass 2M Find the center of mass of the assembly.

2M

L

(50) Find the center of mass of a uniform cone of radius R and height h, see Fig.7.47

(Hint: Divide the cone into an infinite number of disks, each of thickness dx.) (51) A pyramid has a height H and square base area of side L, see Fig.7.48 Find

the center of mass of the pyramid above its base Calculate z for the Great

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Pyramid of Khufu at Giza, Egypt, which has height H = 138.8 m and base square area of side L = 230.4 m (Hint: Divide the pyramid into an infinite number of squares, each of height dz.)

h

x

y

z

R

L

y

z

H

x

Section 7.5 Dynamics of the Center of Mass

(52) The velocities of two particles of masses m1= 2 kg and m2= 3 kg are given

by the position vectors→v1= (4→i + 5→j )(m/s) and →v2 = (2→i − 3→j )(m/s),

respectively Find the velocity of the center of mass of that system

(53) A ball of mass m1= 2 kg traveling with velocity→v1= 15→i m/s collides

head-on and elastically with a sechead-ond ball of mass m1= 3 kg traveling with velocity

→

v2 = −4→i m/s, see Fig.7.49 (a) Find the velocities of the two balls after the collision (b) Find the velocity of the center of mass before and after the collision

2 1

2

m

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224 7 Linear Momentum, Collisions, and Center of Mass

(54) Two particles of masses m1= 0.2 kg and m1= 0.3 kg are initially at rest 2 m

apart The two particles form an isolated system Each particle starts to attract the other with an equal internal constant force of magnitude 0.12 N (a) What

is the speed of their center of mass before and after the start of the attractive

force? (b) What distance does m1move before colliding with m2? (c) What

will be the speed of m1and m2just before the collision?

(55) A man of mass m = 70 kg stands on one end of a flat boat, which is always moving horizontally without friction at a speed ofv◦= 5 m/s over water The

boat has a mass M = 210 kg and length L = 20 m, see Fig.7.50 The man starts

to walk to the other end in the direction of the boat’s motion with a relative speedvrel= 2 m/s (a) What is the location and velocity of the center of mass

before and after the motion of the man? (b) What is the velocityvBof the boat once the man starts to move? (c) What time does the man take to reach the other end, and how far has the boat moved?

M

0

t =

Time t

CM

x

B

CM

x

L

CMt

B

CM

=

°

(56) A projectile is fired from the ground with an initial speedv◦of 40 m/s at an angleθ◦of 15◦m/s above the horizontal direction At the maximum height, the projectile explodes into two fragments of equal mass, see Fig.7.51 One fragment stops momentarily and falls vertically, while the second one flies

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initially in a horizontal direction How far from the ground do the center of

mass and the second fragment land? Take g= 10 m/s2.

Fig 7.51 See Exercise (56) y

x

R

°

θ° Path of CM

Section 7.6 Systems of Variable Mass

(57) A stationary grain funnel drops grain at a rate dM /dt = 840 kg/min onto a

railroad car moving with a constant speedv = 3.5 m/s, see Fig.7.52 (a) What external force must be applied to the car to keep it moving at constant speed (in the absence of friction)? (b) Find the power delivered by this force (c) Find the rate of the kinetic energy acquired by the falling grains

Grains

ext

F

(58) During the first second of its flight in free space, a rocket ejects exhaust that

is 1/50 of its mass with a relative speedvrel = 2.5 × 103m/s What is the

acceleration of the rocket?

(59) A rocket of mass M = 3,000 kg ejects fuel and oxidizer at a rate of 150 kg/s in order to acquire an acceleration a = 4g What is the relative speed of the exhaust

and the thrust on the rocket?

(60) A rocket of mass M = 3,000 kg is moving in free space with a speed

v = 3 × 102m/s relative to the Earth The rocket ejects fuel at a rate of 15 kg/s with a relative speed of 2.5×103m/s (a) Find the thrust on the rocket (b) Find

the final speed of the rocket when its fuel burns completely after 20 s.

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Rotational Motion 8

In this chapter, we first treat the rotation of an extended object about a fixed axis This is commonly known as pure rotational motion The analysis is greatly simplified when the object is rigid To perform this analysis, we first ignore the cause of rotation and describe the rotational motion in terms of angular variables and time This is

known as rotational kinematics We then discuss the causes of rotation This is

known as rotational dynamics , and through the study of this topic we introduce the

concept of torque After that we treat some general cases where the axis of rotation

is not fixed in space In these cases, rigid bodies can undergo both rotational and translational motion, as in the rolling of objects

8.1 Radian Measures

One radian (1 rad) is the angle subtended at the center of a circle of radius r by an

arc of length s equal to the radius of the circle, i.e s = r, see Fig.8.1a Since the

circumference of a circle of radius r is s = 2πr, where π 3.14, then 360◦(or one revolution) corresponds to an angle of (2πr)/r = 2π rad, see also Appendix B.

Thus:

1 rev= 360◦= 2π rad ⇒ 180◦= π rad (8.1)

Therefore:

1◦= (π/180) rad 0.02 rad

1 rad= 180◦/π 57.3◦.

Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-642-23026-4_8,

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Fig 8.1 (a) The definition of

one radian (1 rad) (b) The

definition of angleθ as the

ratio of the arc length s to the

radius r

r r

s=r 1rad

r

r θ s

Generally, ifθ (in radians) represents any arbitrary angle subtended by an arc of length s on the circumference of a circle of radius r, see Fig.8.1b, then the following relation must be satisfied:

θ = s

8.2 Rotational Kinematics; Angular Quantities

Angular Position

The rotational motion of a rigid body (or a particle) about an axis is completely specified by an angleθ that a fixed line in the rigid body (or the particle) makes with some reference fixed line in the space, usually chosen as the x-axis Additionally,

the rotational motion is greatly simplified ifθ is expressed in radians This angle θ

is defined as the angular position of the rigid body (or the particle).

Figure8.2represents a rigid body that is rotating about a fixed axis passing through

point O, where that axis is perpendicular to the plane of the figure Line OP is fixed

in the body and completely specified at time t by the angular position θ which the line OP makes with respect to the x-axis Therefore, the angular position of the rigid body, or the particle at point P which has polar coordinates (r, θ), is:

Angular Displacement

When the rigid body rotates as shown in Fig.8.3, the angular position of the line OP

changes fromθ1at time t1 toθ2at a later time t2 The quantity θ = θ2− θ1is

defined as the angular displacement of the a rigid body:

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