# Work and Energy

## Vector Products

### Vector Dot Multiplication

Vector dot multiplication produces a scalar quantity. This quantity is known as the dot product or scalar product. For two vectors, A and B, oriented at and angle θ to each other, the dot product is defined as shown:

A.B = ABcosθ

Remember that the dot product only has magnitude, not direction. Note that the cosθ term means the dot product is maximum when vectors A and B are parallel (θ=0) and zero when perpendicular (θ=90).

### Vector Cross Multiplication

Vector cross multiplication produces a vector quantity. This quantity is known as the cross product or vector product. For two vectors, A and B, oriented at and angle θ to each other, the cross product's magnitude is given by:

AxB = ABsinθ

The vector product is always directed perpendicular to original vectors (A and B). The direction can be determined using the right hand rule: Thumb x index finger = middle finger. The thumb and index fingers form the vectors A and B, and the index finger points perpendicular to both of them, indicating the direction of the cross product.

## Mechanical Work

Work is when a force acts over a distance. For example, work is done when a crane lifts a pallet of bricks from ground height to the top of a building. If the force (F) is constant, work (W) is the scalar product of the force and the distance though which it acts (s). Work is measured in Newton meters.

W = F.s = Fscosθ

If the force varies, the work is found by integrating over the distance through which it acts.

W = ʃ Fcosθ ds

Examples of varying forces include gravity (over large distances) and springs.

### Example

A good example of work in action is pulleys. Imagine lifting a 20kg block 2m off the ground. This will be hard work!

``` Using F=ma we can see:
F = 20kg * 9.81m/s2 = 196.2N
Then using W=Fscosθ:
W = 196.2N * 2m * cos(0) = 392.4Nm
(note that θ is 0 since the lifting force is in the same direction as the distance we are lifting).
```

So we use a force of 196.2N to do the work. This requires a lot of effort (find something with a similar mass and lift it to see).

Now imagine we use a pulley system so that we pull a rope to lift the block. The pulley system is designed so that we have to pull the rope through 4m to lift the block 2m off the ground. Note carefully that nothing about the block itself has changed - it still has the same mass and we are lifting it the same height, and so the same work is being done. However, we are exerting a force (pulling the rope) over a longer distance. This will affect the magnitude of the force.

``` W = 392.4Nm (same work is being done as before)
F = W/(scosθ) = 392.4 / (4cos0) = 98.1N
```

The force is only 98.1N - half what it was before. Using machines like pulleys makes work 'easier' (less force) at the cost of exerting that force over a greater distance.

## Hooke's Law

The force (Fs) exerted by a spring is proportional to the distance that the spring has been stretched (s). The constant of proportionality is the spring constant (k), which is a property of the spring. The negative sign indicates that the force from the spring is in the opposite direction to the stretching (or compression). A stretched spring will pull inwards (positive s, negative F) and a compressed spring will push outwards (positive F, negative s) as they try to return to their original shape.

Fs = -ks

Note that Hooke's law can be applied to various stretching objects, not just springs. Note: Hooke's law is in fact an approximation, which holds only for small displacements (small s). Larger displacements can cause permanent deformation of the spring, but Hooke's law is sufficient for most applications, including this course.

### Work to deform a spring

Springs are a good example of work done by a changing force. Below is a simple example using integration to find the work required to stretch a spring.

Problem: calculate the work required to stretch an unstretched spring (k=90N/m) by 0.2m.

``` W = ʃ Fcosθ ds
```
• θ is 0 since we assume that the force is acting in the same direction as the spring is stretching.
• To integrate with respect to ds, all non-constant terms inside the integral must be defined in terms of s. (This step is a basic part of most physics integration problems).
• F = ks, so substitute ks in terms of F
• cosθ = cos0 = 1 and is constant (we assume the direction of the force doesn't change)
• Since the spring is initially unstretched, the lower limit of integration will be 0.
• The upper limit of integration will be wherever the stretching stops, in this case 0.2.

Thus we have:

``` W  = 0ʃ0.2 ks ds
= [0.5*ks2]00.2
= 0.5*90*0.22 - 0
= 1.8Nm
```

### Elastic and Inelastic Objects

Elastic objects exhibit Hooke's law and will return to their original shape after deformation. Inelastic objects do not deform (deformation is assumed to be zero since they deform so little). For example, an inelastic string will transmit the entire force applied to it, but an elastic string will stretch, converting part of the force into elastic potential energy.

## Kinetic Energy and the Work-Energy Theorem

All moving objects have kinetic energy. Kinetic energy is defined as half of an object's mass times the object's velocity squared.

K = ½mv2

The work-energy theorem states that the net work done on an object is equal to the change in kenetic energy of that object. This is true as long as the work is only changing the object's speed (not storing potential energy, for example).

## Conservative and non-conservative forces

### Conservative Forces

• A conservative force is one which does no work on a closed path.. This means that is a conservative force is acting on an object, and the object moves around and returns to its starting position, no work will be done by that force.
• For example, imagine pulling the end of a spring and then returning it to its original position. You do work to stretch the spring, and then the spring does work (you do negative work) as it returns to its original size. The positive and negative work cancel out for zero net work.
• The work done by a conservative force is independent of the path taken between two points.
• Imagine two points, A and B, inside a constant gravitational field. The work done by gravity when an object moves from A to B is the same regardless of the route taken. Whether the object moves in a zigzag, goes on a long detour, or travels in a direct line makes no difference.
• Calculating work done by a conservative force is easy because we can just integrate over a straight line from A to B, ignoring the actual path taken.
• Examples of conservative forces include gravity and spring force.

### Non-conservative forces

• Non-convervative forces will do work on a closed path, and the the path taken affects the amount of work done.
• Friction is an example of a non-conservative force.
• Friction always acts opposite to the motion of an object - there is no way to "get back" the work done by friction.

## Potential Energy

Conservative forces are able to "store" potential energy. If work is done against a conservative force, it will be stored as potential energy. Potential energy is measured in Joules.

Wagainst = ΔU = -ʃ Fcosθds

The work done against a conservative force is equal to the increase in potential energy. The integral is negative because the work is in the opposite direction to the force. Note that since the work describes a change in potential energy, the choice of U=0 is arbitrary, and convenient limits should be chosen to fit the circumstances. For springs, U=0 is usually when the spring is unstretched (s=0).

The conservative force may be found by differentiating the potential energy with respect to displacement (when the force is parallel to the displacement):

F = -dU/ds

Non-conservative forces will not store potential energy; instead they change it into other forms (such as heat or sound).

### Elastic Potential Energy

The elastic potential energy is half the spring constant times the displacement squared.

Us = ½ks2

### Gravitational Potential Energy

Gravitational potential energy is the product of mass, gravitational acceleration, and change in height.

ΔUg = mgΔy

## Conservation of Mechanical Energy

Mechanical energy is the sum of kinetic and potential energy.

E = K + U

In some circumstances, the mechanical energy of a system is conserved (remains constant).

When non-conservative forces do work, they change the mechanical energy of a system. For example, friction removes energy from a system by converting kinetic energy into heat and sound, and motors add kinetic energy by converting from electrical energy. Conservative forces, however, do not remove mechanical energy from a system - they simply convert between kinetic and potential energy. If only conservative forces do work, the mechanical energy of a system will not change.

If non-conservative forces do no work, mechanical energy is conserved. In equation form,

ΔE = ΔK + ΔU = 0

The change in mechanical energy is zero. This means that the change in kinetic energy is equal to the negative change in potential energy.

ΔK = -ΔU

Note the emphasis on "change" - the actual kinetic and potential energies are not equal (K≠U).

It is also worth noting that mechanical energy is never conserved in real life - there is always some energy loss. Often these losses are are small, though, and so are ignored for the purposes of making calculations.

### Example

Imagine a 50g ball which bounces on an ideal flat surface, losing no energy to friction or other losses. This means mechanical energy will be conserved. The ball is initially held 3m above the surface. At this point, it has gravitational potential energy:

``` Ug, initial = mgy
= 50*9.81*3 = 1471.5J (taking Ug=0 at the floor)
```

The ball has 1471.5J of gravitational potential energy. It is not moving or compressed, so it has no elastic potential or kinetic energy.

The ball is then released. As it falls, its gravitational potential energy is converted to kinetic energy. When it reaches the floor, all its energy is kinetic. Since mechanical energy is conserved,

``` Kmax = Ug, initial = 1471.5J
```

The ball then strikes the floor and deforms as the kinetic energy is converted into elastic potential energy. When it reaches maximum deformation,

``` Us, max = Kmax = 1471.5J
```

The ball will then begin to return to its original shape and leave the floor as its elastic potential energy is converted into kinetic energy. As soon as it leaves the floor, it will begin storing gravitational potential energy until it reaches its initial position again. Since mechanical energy is conserved, the ball has mechanical energy of 1471.5J throughout the process, and this energy is transferred between kinetic and potential as the ball accelerates and decelerates. Note that the ball's size is ignored for the purposes of this example.

## Power

Power is the rate at which energy is transferred. It is measured in Watts, or joules per second. Work is a form of energy transfer and so power can be described as the rate at which work is done.

Instantaneous power is the rate of work (energy transfer) at any single point in time. It is found by differentiating the work with respect to time.

P = dE/dt = dW/dt

Average power is the average rate of work done (W) over a time period (Δt).

Pavg = W/Δt

## End

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