Ch 8 Potential energy and Conservation of Energy. Question: 2, 3, 8, 9 Problems: 3, 9, 15, 21, 24, 25, 31, 32, 35, 41, 43, 47, 49, 53, 55, 63

Ch 8

Potential energy

and Conservation of Energy

Question: 2, 3, 8, 9

Problems: 3, 9, 15, 21, 24, 25, 31, 32, 35, 41, 43, 47, 49, 53, 55, 63

Potential energy

Kinetic energy – energy due to motion

Potential energy – energy due to the arrangement of the objects in a system (U)

Potential energy is due to an objects location.

gravitational potential energy – due to the separation between two objects

elastic potential energy – energy associated with the compression or stretching of an elastic object

Potential energy is a form of energy so the units are Joules.

Potential energy is energy that can be stored and later released to produce motion.

Examples:

Shove a ball into a spring loaded gun. The elastic potential energy is stored and when you shoot the ball the potential energy is converted to kinetic energy.

After climbing a tree you have gravitational potential energy stored up. Fall out of the tree that potential energy is converted to kinetic energy as you fall down.

Work and potential energy

Throw ball vertically upward.

As the ball rises, gravity does negative work on the ball.

The ball slowed down (lost kinetic energy).

The kinetic energy is transferred to the gravitational potential energy of the ball-Earth system.

As the ball is falling, gravity does positive work on the ball.

The ball speeds up (gains kinetic energy).

The kinetic energy comes from the gravitational potential energy of the ball-Earth system.

As ball rises: K < 0

When the gravitational force did negative work, the GPE increased, U > 0.

As ball falls: K > 0

When the gravitational force did positive work, the GPE decreased, U < 0.

From chapter 7: W = K W = - U

Block hitting spring

Replace throwing a ball upward with a block sliding into a spring that is fixed to a wall.

When the block hits the spring the block has kinetic energy.

As the block compresses the spring, it loses kinetic energy and spring gets elastic potential energy.

When block comes to rest, all the kinetic energy has been turned into potential energy.

As the spring expands the transfer of energy is reversed from potential energy to kinetic energy.

Conservative forces

Assume a force acts between a system of 2 objects.

As the force changes the configuration of the system, the force does work, W₁, energy is transferred from kinetic energy to

some other form of energy.

When the change in configuration is reversed, the force does work, W₂, and the transfer of energy is reversed.

If W₁ = -W₂ is true the other type of energy was potential energy.

The force is a conservative force.

examples: of conservative forces: gravity, springs, electric force

If W₁ = -W₂ is not always true, the force is nonconservative.

Examples of nonconservative forces: friction, drag

Nonconservative forces convert kinetic energy to thermal energy. Transfers to thermal energy cannot be reversed.

Example: Kinetic energy can be lost to friction, but friction cannot be used to produce kinetic energy.

Conservative forces are path independent.

The work done by a conservative force only depends on the endpoint.

How you get from the initial configuration to the final configuration does not matter.

The net work done by a conservative force around a closed loop is zero.

Example: Throw a ball up and let it fall back down. The total work done by gravity is zero.

Since conservative forces are path independent, we usually only care about the endpoints. This can make a complicated problem easier to solve.

Nonconservative forces are path dependent.

When dealing with nonconservative forces, such as friction, we need to consider the path the object takes.

|W_f| = F_f d where d is the path length.

Drag a box over a rough surface from point 1 to point 2. Friction will do more work when using path A than path B.

If you dragged the box along A from 1 to 2 then dragged it back using path B the total work done by friction is not zero.

A

B 1

2

How to calculate values of potential energy.

In general, the work done by a force is:

Since U = -W

Gravitational Potential Energy (picking up to be positive)

dx x

F

W ^f

i

x

x ( )

dx x

F

U ^f

i

x

x ( )

y mg U

y y

mg dy

mg dy

mg

U ^f

i f

i

y

y f i

y

y ( ) ( )

Gravitational Potential Energy

Change in GPE from one point to another is:

Only changes in GPE are physically meaningful.

U – U_i = mg(y – y_i)

We will often compare GPE to the GPE at a reference point.

Usually we use: U_i = 0 at y_i = 0.

So: U(y) = mgy

You can set any height to be the zero of GPE. This will be a big convenience in solving problems.

y mg U

y y

mg dy

mg dy

mg

U ^f

i f

i

y

y f i

y

y ( ) ( )

Elastic potential energy (EPE)

When spring is relaxed, x_i = 0, the U_i = 0 U – 0 = ½ kx² – 0

U(x) = ½ kx²

Remember from last chapter W_s = - ½ kx² So this fits U = -W

f

i

f

i

x x

x

x xdx kxf kxi

k dx

kx

U ^{2 2}

2 1 2

) 1 (

Conservation of mechanical energy

Mechanical energy: E_mech = K + U

If you only have conservative forces, the total mechanical energy never changes.

K₂ + U₂ = K₁ + U₁ and

E_mech = K + U = 0

E_mech = K + U = 0

This is definitely one of the most important and most useful rules in physics.

If there are only conservative forces involved, and there are multiple steps, you can ignore the intermediate configurations of the system.

Will see a good example of this later.

Potential Energy Curves

Graph of potential energy vs. position See fig. 8-9

For a particle moving in the x-direction U(x) = -W = -F(x) x

The derivative (slope) of the potential energy curve is related to the force.

dx x dU x

x

F U( ) ( )

The force doing the work, is the negative of the slope in the potential energy curve.

Example: spring U(x) ½ kx²

kx dx kx

d dx

x x dU

F )

2 (1 )

) (

( ²

U(x)

x

U(x) + K(x) = E_mech K(x) = E_mech – U(x)

Where U(x) = E_mech the kinetic energy will be zero.

This is a turning point.

Think of the potential energy curve as a rollercoaster track.

The height of the track is U(x). When the cart reach a point where its total energy is U(x) , the cart stops and turns around.

In Newtonian (classical physics), particles are confined between turning points.

Equilibrium points.

Points on the potential energy curve are said to be either unstable or stable.

A point is unstable is a small displacement in either direction will lead to a greater

displacement. (relative maxima)

A point is stable is after a small displacement in either direction, the particle will return to its original position. (relative minima)

Work done by an external force.

If positive work is done by an external force on a system, energy is transferred to the system.

Negative work done on a system results in energy transferred from the system.

W = U + E = E_mech

Friction

Look at case where a force, F, dragged a box across a floor.

Newton’s 2^nd law gives: F – f_k = ma

a is constant so:

d f K

Fd

d f mv

mv Fd

d v a v

ad v

v

f ma F

k

k o

o

k

2 2

2 0 2

2 2

2 1 2

1 2

) (

2

The friction between the two surfaces causes their temperatures to rise.

The sliding increases the thermal energy by an amount equal in magnitude to the work done by friction: E_th = f_kd

Fd = E_mech + f_kd = E_mech + E_th

Work done on a system when there is friction:

W = E_mech + E_th

Conservation of Energy

The total energy of a system can change only by amounts of energy that are transferred to or from the system.

Energies to be considered are mechanic, thermal, and internal

When work is done, the work is equal to the sum of the changes in these types of energies.

W = E = E_mech + E_th + E_int

Conservation of energy for an isolated system:

E_mech + E_th + e_int = 0

For an isolated system, the total energy cannot change.

Power:

Average power is the average rate that a force transfers energy from, one type to another.

Instantaneous power:

Problems: 18, 26, 34, 44, 46, 58

t P_ave E

dt P dE