In this lecture the following are introduced:
• The work done by a force
• Conservative forces
• Kinetic energy
• Potential energy
• Conservation of mechanical energy

Momentum and Vis Viva

An argument developed after Newton proposed momentum (the product of mass and speed) as the fundamental quantity of motion. Leibniz (1646-1716) championed the idea that the product of mass and the square of speed ("vis viva" or living force) was the fundamental quantity of motion.

The argument continued for the following 150 years or so. We will see how it was resolved after a few concepts have been developed.

We have seen that the effect of force through time is to change the momentum (lecture 16). It is now time to look at the effect of force through space.

The effect of force through space.

Here we see a force, F, pushing at an angle downwards onto an object placed on a horizontal plane. The object cannot go into the plane and is forced along the plane as shown by the displacement vector r.

Only a part of the force causes the motion. We can work out which part from a vector analysis.

The work done by a force, is defined as the size of the actual force which causes the displacement multiplied by the size of the displacement. The size of the force that causes the displacement is that part of the force which is in the same direction as the displacement, i.e. Fcosθ. The work done by the force is then Fcosθ·r.

Notice that both force and displacement are vectors. However, since we are only concerned with multiplying how much of the force is in the direction of the displacement, the direction does not matter and the result is a scalar quantity.

To represent the product of two vectors lining up to give a scalar, we use a dot for the "Cos" function.

The unit of work done by a force is the Joule.
1 Joule of work is done when a force of 1 Newton displaces an object by 1 metre.

Example
A 5 kg mass is moved from point A, vertically upwards through 3 m to point B, then upwards at 30⁰ to the horizontal through 2 m to point C, then horizontally through 4 m to point D. Find the work done by gravity in moving from A to D.

From A to B The minus sign says that the force of gravity did a negative amount of work, i.e. some other force did work against gravity to move the mass.

From B to C Again, some other force did work against gravity to move the mass.

From C to D Horizontal movement does not involve work by or against gravity. Only vertical movement, which is directly with or against the force of gravity, will see work done by or against gravity.

One fascinating feature is that it does not matter what path is taken from A to D, the same negative amount of work is done by gravity. However, the external agent that works against gravity, will be affected by the path.
Conclusion: In travelling a closed path from A to D and then back to A again, the net work done by gravity is zero. The work done against gravity in going up, is matched by the work done by gravity in going down. When this happens, the force is said to be a conservative force.

Conservative forces

A conservative force does no net work in going through a closed path. Gravity is a conservative force. Friction is a non-conservative force because it always opposes motion, it never adds to it.

Resolving the 150 year argument.

The argument was resolved when the quantities could be expressed properly. We start with the effect of force through distance.

The switch in the third line was the hard bit, now what does the momentum term mean?

The effect of momentum through speed is the shaded area

The quantity ½m·v² is called the kinetic energy (symbol T), and has the Joule for its unit as it is still a force times a distance.

Thus, the effect of force through distance, which is called the work done by the force, changes the kinetic energy. Whereas, the effect of force through time, which is called the impulse of the force, changes the momentum.

What happened was that one lot of scientists did measurements of force and time, ending up with momentum.
Another lot did measurements of force and distance, ending up with kinetic energy (really its near equivalent "vis viva").
This has considerable interest for kinematics, because it makes questions involving speed and distance easier.

Example
A child pulls a toy car of mass 12.5 kg along a level floor. The force exerted is 10 N upwards at 20° to the horizontal and the car moves a distance of 6 m. Find
(a) the work done by the child, and
(b) the speed of the car after the 6 m.

The work done by the child = 10Cos20o × 6 = 56.4 J.

The long way is with Newton's law, and goes like this:

The short way, from work/energy, goes: The work done by the child = increase in kinetic energy

Example
A 200 kg swordfish swimming at 5 m.s^-1 rams a wooden yacht, and the fish is stopped in 0.8 m. Find
(a) the initial kinetic energy of the fish, and
(b) the average force exerted by the wood on the swordfish.

The force of the wood on the swordfish was a reaction force which equaled the force applied by the swordfish.

Potential Energy

When work is done against a conservative force, such as lifting a mass against gravity, the system then has the ability to recover that work, by for example, letting the mass fall. The ability to recover work is a measure of the potential energy of the system. Be careful to distinguish the work done by a force, and the work done on (or against) a force.

Definition
The change in potential energy of a system
  = the work done against the system force (by some external agent)
  = the negative of the work done by the system force
Potential energy will also have the Joule as its unit.

Gravitational Potential Energy

Near the surface of the earth the acceleration of gravity is taken to be constant at g=9.8 m.s^-2.
The work done by gravity in lifting a mass, m, a vertical distance, h, is given by:

The gain in potential energy = the negative of work done by gravity = +mgh.

Mechanical Energy Conservation

Mechanical work or energy is defined to be the sum of potential and kinetic energy. Since potential energy is only defined for conservative forces (which do not dissipate energy), potential energy can be changed without loss into kinetic energy by the system force doing its non-dissipative work. Thus in the absence of dissipative forces mechanical energy is conserved when transforming between potential and kinetic energy.

Example

An object of mass, m, falls from rest at a height, h, under gravity. Find its speed just before it hits the ground.

Example
A mass of 20 kg slides down a smooth plane inclined at 37° to the horizontal and the force of gravity moves the mass through 10 m. Find
(a) the work done by gravity, and.
(b) the speed of the mass after the 10 m.

Rider: However, there are usually energy dissipating forces and effects in energy changes, e.g. friction. In general, when energy changes from one form to another, there is some energy lost.

Summarising:

The work done by a force, W = F · r = F·Cosθ·r in Joule.
A conservative force does no net work in going through a closed path.
The kinetic energy, T = ½m·v² in Joule.

The change in potential energy of a system	= the work done against the system force
	= the negative of the work done by the system force

Gravitational potential energy (near the surface of the Earth) = mgh.
In the absence of dissipative forces mechanical energy (i.e. KE and PE) is conserved.