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 changes the momentum.
We now look at the effect of force through space.

The effect of force through space.

Here we see a force, F, pushing down at an angle to an object on a horizontal plane. The object cannot go into the plane and is forced along the plane as shown by r. Only part of the force causes motion. We can work out which part causes motion from vector analysis.

The angle between the vectors is θ. (Put the two tails together). The component of the force which tries to push the object into the plane, is Fsinθ. This creates the normal reaction of the plane upwards on the object. This component, and the normal reaction, form internal forces to keep the object in contact with the plane. The component of the force which pushes the object in the same direction as the plane, is Fcosθ. This is the external force which moves the object along the plane.

W, the work done by the force, is defined as the effect of the force which moves the object (Fcosθ) along the length of the displacement (r), i.e. the product

Notice that both force and displacement are vectors, but since we are only concerned with how much they are in the same direction as each other, (the "cosine" does that) the actual product of the two vectors is a scalar. This is called a scalar product of vectors.

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

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 of path independence

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 upwards.
From B to C	Again, work was done against gravity in lifting the mass upwards.
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.

The work done by gravity in moving from A to D is -147 - 49 + 0 = -196 Joule. The total vertical height moved was 4 m. To move the weight 5g N directly through 4 vertical metres is also -196 Joule.
It can be concluded 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 conditions of the path. This means such things as friction, viscous drag etc.

Another conclusion, is that by travelling a closed path from A to D and then back to A again, the net work done by gravity will be zero. That is, the work done against gravity in going up, is matched by the work done by gravity in going down. If when moving through any closed loop there is no net work done by or against a force, that force is said to be a conservative force.

Conservative and Non-conservative Forces

Another conclusion from the example is, that by travelling the closed path from A to D and then back to A again, the net work done by gravity will be zero. That is, the work done against gravity in going up, is matched by the work done by gravity in going down. When in moving through a closed loop, a force does no net work, the force is said to be a conservative force.

A conservative force does no net work in going through a closed path.

Friction is a non-conservative force because it always opposes motion and never encourages it. The amount of work done against friction depends explicitly on the total length and conditions of the path.

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 of "dt" from under "dv" to under "dr" in the third line was the hard bit to conceive.	Now what does the effect of momentum through velocity mean?
With the mass constant, momentum increases linearly with time as shown in the graph.	The triangular area under the graph This area gives the change in a quantity called Kinetic Energy, T, defined as: Kinetic Energy T = ½ mv2.

The effect of Force through space is to change the Kinetic Energy ½ mv².
Kinetic energy has the same unit as Work, i.e. the Joule.

Resolving the argument: there are two fundamental quantities of motion, Momentum and Kinetic Energy.
The effect of force through time, (called the Impulse of the force) changes the Momentum.
The effect of force through space, (called the Work done by the force) changes the Kinetic energy.

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").

Kinetic energy has considerable interest for kinematics, because it makes questions involving speed and distance easier.

Example W1

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 horizontally 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 force which moves the car is 10cos200. The work done by the child is F·r Joule.
The long way to find the speed is from Newton's law, and goes like this:	The short way, from work/energy, goes like this: Work done by child = increase in K.E.

Example W2

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 will be a reaction force to the force applied by the swordfish, i.e. 3125 N in the opposite direction to the swordfish's motion.

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, for example, in the case of gravity by letting the mass fall. The ability to recover work is a measure of the potential energy of the system. Work done by a force actually moves a object through a distance, and potential energy is the unrealised ability to move an object through a distance. There will be as many types of potential energy as there are types of conservative force.

Definition

Potential energy will also have the Joule as its unit.
It has been seen that force is determined by how quickly the momentum changes in time, and later it will be seen that force is also determined by how quickly the potential energy changes with distance.

Gravitational Potential Energy

As gravity is a conservative force, there will be gravitational potential energy. Near the surface of the earth the acceleration of gravity is taken to be downwards at a constant value of g = 9.8 m.s^-2.

The work done by gravity in lifting a mass, m, upwards from the surface of the Earth by a vertical distance, h, is given by:

W_g = F_g·r = mg×cos180⁰×h = - mgh J.
The angle is 180⁰ because the force of gravity acts downwards and the displacement of the mass is upwards.
The change in the Potential Energy of the system is the Work done against gravity, or the negative of the Work done by gravity = + mgh. The Potential Energy changed from zero to +mgh.

Mechanical Energy Conservation

Mechanical work or energy is defined to be the sum of potential and kinetic energy of a system.

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 mechanical energy is conserved in the absence of dissipative forces.

Example W3

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

Example W4

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.

The angle between the force of gravity and the displacement is 530 (put the tails together).

Summarising:

The work done by a force is W = F·r Joule, i.e. Fcosθ.r, the moving force times the distance.

The kinetic energy, T =  in Joule.

A conservative force does no net work in going through a closed path.

Gravitational potential energy = mgh.

In the absence of dissipative forces mechanical energy (i.e. the sum of KE and PE) is conserved.

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