Work and Energy

Conservative/Non-conservative Forces

Important Concepts:

• The total amount of energy in the Universe is conserved. This is a wonderful property of our Universe: Energy can be transformed from one type to another and transferred from one object to another, but the total amount is always the same. This is the principle of conservation of energy. No exception to this rule has ever been found.

• The total amount of energy within any system is conserved, as long as no energy enters or leaves the system. It remains constant as long as the same amount of energy is added and removed.

• Kinetic energy, K, is associated with the state of motion of an object. When an object is stationary, its kinetic energy is zero.

21

2K mv

Potential energy, U, is energy associated with an object’s position relative to another object or

the arrangement of a system of objects.• Gravitational potential energy, UG, is the potential energy of

an object or system due to interactions of gravitational fields.

GU mgh

G

GmMU

r

near the surface

“farther afield”

*Why is it negative? Because U is 0 at infinity and gets larger and larger negative as a mass moves closer to Earth.

Want a more complicated answer? F=-U, and the gravitational force is negative.

• Elastic potential energy is described for a system such as a spring that has a defined spring constant, k, that is extended a distance x from a position of equilibrium.

21

2

s

s

F kx

U kx

• Electric potential energy is described for a charged particle q near another charged particle, Q: E

kqQU

r

Mechanical energy includes kinetic energy and potential energy.

• In the case of a falling object (neglecting air friction), the gravitational potential energy is converted to kinetic energy as the object falls—maintaining constant total mechanical energy.

• If we do include air friction, the total mechanical energy is not constant as the object falls, because some of the mechanical energy has been converted to thermal energy in the molecules of the object and the air (which means a temperature increase in both the object and the air).

The work done by any force on an object or system is:

W F s

•Work is a scalar quantity, measured in Joules, that is positive if the force and displacement are in the same direction and negative if the force and displacement are in opposite directions.

•Positive work done on a system increases the mechanical energy of the system.

•Negative work done on a system decreases the mechanical energy of the system.

Example: As a ball rolls across a floor, its gravitational potential energy does not change, but its kinetic energy decreases as the ball rolls across the floor. Explain this in terms of conservation of energy.

The friction force on the ball as it rolls is in a direction opposite the direction that the ball rolls. Thus, the dot product of the friction force and the displacement is negative. The friction force does negative work on the ball, eventually decreases its kinetic energy to zero as it converts that mechanical energy to thermal energy.

A box of mass m is pushed by a force F to the top of a ramp of length d and height h. Determine the work done by the force in pushing the box to the top of the ramp (a) neglecting friction, and (b) including friction, with coefficient .

h d F

First, add the forces on the box. Then use d and h to determine the angle .

F

N

mg

Construct the components for mg.

F

N

mg

mgcos

mgsin

F

N

mg

mgcos

mgsin

Method 1: Energy method for determining work with no friction Work = Change in Potential Energy W = U = mgh

Method 2: Force method for determining work with no friction. W = Fs where F = mgsin and s = d W = mgsind BUT dsin = ______ SO: W = _______

F

N

mg

mgcos

mgsin

Now we consider the same situation with friction.

F

N

mg

mgcos

mgsin

Ff

Method 1: Energy method with friction W = Change in Potential Energy + Loss to Thermal Energy due to Friction

W = mgh + FfdW = mgh + mgcos

F

N

mg

mgcos

mgsin

Ff

Method 2: Force method of determining work with friction

W= F s where F = Ff + mgsin and s = dW = mgcos d + mgsin d

W = ____ + _______

F

N

mg

mgcos

mgsin

Ff

A Combination Pendulum Motion

A pendulum of length L is released from angle . When it swings to vertical, it hits a rod that is perpendicular to the plane of the swing (i.e., it projects out of the page) and positioned at ½ L.

Find the angle to which the pendulum will swing after hitting the bar.

?

Consider the initial potential energy of the pendulum bob and the final kinetic energy of the bob.

mgL(1 - cos) = mg(1/2 L)(1 - cos ) 1 - cos = (1/2) (1 - cos )

½ = cos - ½ cos