Work and Energy

Work Done by a Constant Force

Definition of work:

The work done by a constant force acting on an object is equal to the the displacement times the component of the force parallel to that displacement.

Units of work: newton•meter (N•m).

1 N•m is called 1 joule.

Work Done by a Constant Force

In (a), there is a force but no displacement: no work is done. In (b), the force is parallel to the displacement, and in (c) the force is at an angle to the displacement.

Work Done by a Constant ForceIf there is more than one force acting on an object, it is useful to define the net work:

The total, or net, work is defined as the work done by all the forces acting on the object, or the scalar sum of all those quantities of work.

Work Done by a Variable Force

The force exerted by a spring varies linearly with the displacement:

Work Done by a Variable Force

A plot of force versus displacement allows us to calculate the work done:

Work is Energy : Kinetic Energy

We know that the net force acting on an object causes the object to accelerate, changing its velocity:

We can use this relation to calculate the work done:

Work is Energy : Kinetic Energy

Kinetic energy is defined:

The net work on an object changes its kinetic energy.

Work is Energy : Kinetic Energy

This relationship is called the work-energy theorem:

EnergyIn physics, energy is an indirectly measured quantity. Energy is a scalar, so many calculations are made very easy when considering energy (no components).

Energy can take many forms. Besides converting work into kinetic energy, we can do work on an object and change its potential energy, usually called U.

W = ΔU = U-U0

Drawing back on a bow does work on the system, this work results in potential energy

EnergyEnergy and mass are related. Like mass, energy cannot be destroyed. A closed system conserves energy, and energy conservation is one of the few fundamental laws of physics.

Potential energy in the bow is converted to kinetic energy in the arrow. Energy can be converted from

one form to another, but it is never “lost”

Potential EnergyPotential energy may be thought of as stored work, such as in a compressed spring or an object at some height above the ground.

We saw earlier the work done when we compress a spring, now we see it goes into potential energy:

Potential energy of a spring:

Potential Energy

Gravitational potential energy

Moving a can up by a distance “y” involves work, the work done is W = F d

W = mgy

U = mgy

Potential EnergyWhere is potential 0? Only changes in potential energy are physically significant; therefore the point where U = 0 may be chosen for convenience.

Conservation of Energy

We observe that, once all forms of energy are accounted for, the total energy of an isolated system does not change.

The total energy of an isolated system is always conserved.

We define a conservative force:

A force is conservative if the work done by it in moving an object is independent of the object’s path.

A force is said to be nonconservative if the work done by it in moving an object does depend on the object’s path.

Conservation of Energy

So, what types of forces are conservative?

Conservative forces:

Gravity is one example; the work done by gravity only depends on the difference between the initial and final height, and not on the path between them.

Nonconservative forces:

One example of a non-conservative force is friction. Why?

Conservation of Energy

Another way of describing a conservative force:

A force is conservative if the work done by it in moving an object through a round trip is zero.

A force is conservative if the path chosen for moving an object has no effect on the final energy

A

Conservation of EnergyWe define the total mechanical energy:

For a conservative force:

Conservation of EnergyExample of how energy considerations can make problems easy: Assume a man throws three balls at 3 angles (+45°, 0°, -45°) from a mountain, with the same speed. What are differences in speeds of the balls just s they hit the ground?

All three of these balls have the same kinetic and potential energy when they start.

Energy is conserved

So, for all three balls, their speeds just when they hit the bottom are exactly the same

Example problem:

Man falls from roof, 5 meters high. How fast is he going when he makes contact with ground?

Choose coordinates so potentialenergy is U on the ground, then

Same answer as before

E1=E2

Conservation of EnergyIn a conservative system, the total mechanical energy does not change, but the split between kinetic and potential energy does.

A look at the Pendulum:

Non-Conservation of EnergyIf a non-conservative force is present, the work done by the non-conservative force is equal to the change in the total mechanical energy.

Where does the energy go? Depends. For friction, heat into the environment (rub hands to see the effect)

Consequence of non-conservative forces: efficiency

Mechanical efficiency:

The efficiency of any real system is always less than 100%, because there always some non-conservative forces to be dealt with.

Example: car engine: energy goes into heat from friction, sound waves into the air,…

Consequence of non-conservative forces: efficiency

PowerPower is defined as energy/time

The average power is the total amount of work done divided by the time taken to do the work.

“High Power” means large energy or short amount of time, or both.

The energy in a lithium battery (laptop) is about the same as the energy in a hand grenade. The difference is their power, the time over which this energy is released.

Δt=10-5s Δt=104s