General physics a1 week 4 work – mechanical energy

❑ Gravitational Potential Energy❑ Elastic Spring Potential Energy ❑ Conservative and Nonconservative Forces ❑ Conservation of Energy... ❑ Gravitational Potential Energy❑ Elastic Spring P

General Physics A1

Week 4: Work – Mechanical Energy

Vietnam National University Ho Chi Minh City

Ho Chi Minh City University of Technology

❑ Gravitational Potential Energy

❑ Elastic (Spring) Potential Energy

❑ Conservative and Nonconservative Forces

❑ Conservation of Energy

Scalar Product of Two Vectors

The scalar product of two vectors is written as

 It is also called the dot product

 θ is the angle between A and B

Scalar Product is a Scalar

Not a vector

May be positive, negative, or zero

Scalar Product: An Example

❑ The vectors:

❑ Determine the scalar product:

❑ Find the angle θ between these two vectors:

ˆ2ˆand

ˆ3ˆ





46-22

3(-1)

60 65

4 cos

65

4 5

13

4 cos

5 2

) 1 (

13 3

2

1

2 2 2

2 2

2 2

B B B

A A

❑ Gravitational Potential Energy

❑ Elastic (Spring) Potential Energy

❑ Conservative and Nonconservative Forces

❑ Conservation of Energy

Definition of Work W

 The work, W , done by a constant force on an object is defined as the scalar (dot) product of the component of the force along the direction of displacement and the magnitude of the displacement

 is the magnitude of the force

 is the the object s displacement ’

 Φ is the angle between and

 N • m = J

 J = ( kg • m / s ) • m 2

Work: Positive or Negative

of the work depends on the direction of the force relative to the displacement

Example: When Work is Zero

Special Case: Constant Acceleration

❑ Gravitational Potential Energy

❑ Elastic (Spring) Potential Energy

❑ Conservative and Nonconservative Forces

❑ Conservation of Energy

Kinetic Energy

 For an object m moving with

a speed of v

 Kinetic Energy is energy

associated with the state of

motion of an object

 SI unit: joule (J)

1 joule = 1 J = 1 kg m2/s2

 When work is done by a net force on an object and the only change in the object is its speed, the work done is equal to the change in the

object’s kinetic energy

 Speed will increase if work is positive

 Speed will decrease if work is negative

Work-Energy Theorem

Wtot K2 K1 K

 On a graph of force as a function

of position, the total work done

by the force is represented by the

area under the curve between

the initial and the final position

 Note there could be negative work!

 Straight-line motion

 Motion along a curve

Work with Varying Forces

F dl

dv v

dt

dx dx

dv dt

dv

dx dx

dv mv

dx ma dx

F

x

x x

mv

Wtot   2  

1

2 2

2

1 2

1

x

v

v xtot mv dv

1

Work-Energy with Varying Forces

forces as well as for constant ones

 Involves the spring constant, k

 Where is the force exerted on the spring in the same direction of x

 The force exerted by the spring is

 k depends on how the spring is made of Unit: N/m.

Spring Force: a Varying Force

 To stretch a spring, we

must do work

 We apply equal and

opposite forces to the

ends of the spring and

gradually increase the

forces

 The work we must do to

stretch the spring from

x1 to x2

Work Done on a Spring

2 1

2 2

2

1 2

1 2

1 2

1

kx kx

dx kx dx

❑ Gravitational Potential Energy

❑ Elastic (Spring) Potential Energy

❑ Conservative and Nonconservative Forces

❑ Conservation of Energy

 Work does not depend on time interval

in the design and use of practical device

Power

 Power is the time rate of energy transfer Power is valid for any means of energy transfer

Instaneous Power

Units of Power

The SI unit of power is called the watt

A unit of power in the US Customary system

❑ Gravitational Potential Energy

❑ Elastic (Spring) Potential Energy

❑ Conservative and Nonconservative Forces

❑ Conservation of Energy

Work Done by Gravity and Gravitational Potential Energy

Wgrav Ugrav,1 Ugrav,2 Ugrav,2 Ugrav,1 Ugrav

Potential Energy

the position of the object

energy associated with the relative

position of an object in space near

 m is the mass of an object

 g is the acceleration of gravity

 y is the vertical position of the mass relative the surface

of the Earth

 SI unit: joule (J)

Reference Level

 A location where the gravitational potential

energy is zero must be chosen for each problem

potential energy is the important quantity

reference height

 often the Earth’s surface

 may be some other point suggested by the problem

for the entire problem

Quiz: Reference Level

 The gravitational potential energy of an object

(a) is always positive

(b) is always negative

(c) never equals to zero

(d) can be negative or positive

Quiz: Reference Level

 The gravitational potential energy of an object

(a) is always positive

(b) is always negative

(c) never equals to zero

(d) can be negative or positive

Recall: Work-Kinetic Energy Theorem

Wtot K2 K1 K

the only change in the object is its speed, the work

energy

 The work-kinetic energy theorem can be extended

to include gravitational potential energy:

done by all rest forces are zero, then

Extended Work-Energy Theorem with

Gravitational Potential Energy

Wgrav Ugrav,1 Ugrav,2 Ugrav,2 Ugrav,1 Ugrav

 We denote the total mechanical energy by

Problem-Solving Strategy

energy

 Do not change this location while solving the problem

between

 One point should be where information is given

 The other point should be where you want to find out something

Prof Water Lewin’s Pendulum

 The work-kinetic energy theorem can be

extended to include potential energy:

 Since

 Then

 If forces other than gravity do work

When Forces other than Gravity Do Work

Wtot K2 K1 K

Wtot  Wgrav Wother  K2  K1

Wgrav Ugrav,1 Ugrav,2 Ugrav,2 Ugrav,1 Ugrav

Wother Ugrav,1 Ugrav,2  K2  K1

K1 Ugrav,1 Wother  K2 Ugrav,21

2 mv1

2  mgy1 Wother  1

2 mv2

2  mgy2

❑ Gravitational Potential Energy

❑ Elastic (Spring) Potential Energy

❑ Conservative and Nonconservative Forces

❑ Conservation of Energy

 Hooke’s Law gives the force

Where is the force exerted on the

spring in the same direction of x

 Work done on the spring from x to x1 2

 The force exerted by the spring is

Spring Force: An Elastic Force

2 1

2 2

2

12

dxkxdx

 Work done by the spring

 The work-kinetic energy theorem can be extended to include elastic potential energy:

all rest forces are zero, then

Extended Work-Energy Theorem with

Elastic Potential Energy

Mechanical Energy Conservation with BOTH Gravitational and Elastic Potential Energy

❑ We denote the total mechanical energy:

❑ Since

❑ The total mechanical energy is conserved:

Exercise: A Block Projected up an Incline

 A 0.5-kg block rests on a horizontal, frictionless surface The block is pressed back against a

spring having a constant of k = 625 N/m,

compressing the spring by 10.0 cm to point A

Then the block is released.

 (a) Find the maximum distance d the block

travels up the frictionless incline if θ = 30 °

 (b) How fast is the block going when halfway to its maximum height?

 Point A (initial state):

 Point C (final state):

Exercise: A Block Projected up an Incline

 Point A (initial state):

 Point B (final state):

Exercise: A Block Projected up an Incline

Types of Forces

 Work and energy associated

with the force can be recovered

 Examples: Gravity, Spring Force,

EM forces

 The forces are generally

dissipative and work done

against it cannot easily be

recovered

 Examples: Kinetic friction, air

drag forces, normal forces,

tension forces, applied forces …

❑ Gravitational Potential Energy

❑ Elastic (Spring) Potential Energy

❑ Conservative and Nonconservative Forces

❑ Conservation of Energy

Conservation of Energy in General

 Any work done by conservative forces can be accounted for by changes in potential energy

 Law of conservation of energy

 Energy is never created or destroyed It only

changes form

 Define the system to see if it includes non-conservative forces (especially friction, drag force …)

 Without non-conservative forces

 With non-conservative forces

 Select the location of zero potential energy

 Do not change this location while solving the problem

 Identify two points the object of interest moves between

 One point should be where information is given

 The other point should be where you want to find out

something

Prolem-Solving Strategy

Roller Coasters

Exercise: Skateboarding

frictionless ramp He moves through a quarter-circle with radius R=3m The boy and his skateboard have

a total mass of 25 kg

Find his speed at the bottom of the ramp

0

0

2 ,

2 2 2

1

2

1 , 1

K

mgR U

K

s m gR

v2  2  7 67 /

02

1

0 mgR  mv22 

2

2 2 1

2 1

2

1 2

1

mgy mv

mgy

Exercise: Skateboarding