# Work & Energy

## 6. Work and Energy

### 6.1 Mechanical work

• When a constant force moves an object in the direction of the force, the work done equals the product of the force and the distance the object is moved in the direction of the force. W = F d Work represents a transfer of energy and therefore has the same units as energy, the joule (J). Like all forms of energy, work is a scalar quantity.
• When an object moves at right angles to the direction of the force acting on the object (e.g. gravity acting on a space shuttle in orbit) then no work is done by that force since the object does not move at all in the direction of the force.

### 6.2 Mechanical work as a product of vectors

• In general, for all possible angles, the work done is equal to the dot product of the force and displacement vectors.
• The dot product of any two vectors is equal to the magnitude of the first vector times the magnitude of the second vector times the cosine of the angle between them. Thus if the vectors are perpendicular then the dot product is zero, as stated above, and if the vectors point in the same direction then the work is given by the product of the magnitudes of the two vectors. If the vectors are antiparallel (i.e. the displacement is in the opposite direction to the force) then the force does negative work is negative since the cosine of 180° is -1. In this case it can be said that work is done ON the force.
• The dot product of any two vectors is itself a scalar quantity.
• Vectors can also be multiplied together to give a 'cross product' which is itself a vector but this will not be explained here.

### 6.4 Kinetic and potential energies

• Energy is the ability to do work.
• Kinetic energy (motion energy) = ½ mass x speed2 K = ½ mv2
• Potential energy is stored energy, often related to the object's position
• Gravitational potential energy = mass x acceleration of free fall x height U = mgh
• Elastic potential energy (energy stored in a spring) = U = ½ kx2 where x is the extension and k is the spring constant (see next line)
• The spring constant k is the ratio of the extending force to the extension k = -F/x The minus sign is there because F is taken to be the force of the spring (not on the spring) and that is in the opposite direction to the extension of the spring.
• To avoid ambiguity, it is acceptable to distinguish between gravitational potential energy and elastic potential energy by using subscripts: Ugrav, Uel
• The sum of the kinetic energy and potential energy is called the 'mechanical energy'

### 6.7 The conservation of mechanical energy

• The law of conservation of energy states that energy cannot be created or destroyed, though it can be transformed from one form to another.
• Note: The above law is useful but imperfect. Einstein pointed out that since energy can in fact be 'created' from mass by converting mass into energy in accordance with his equation E = mc2 (c is the speed of light, 3 x108 m/s). The quantity that is really conserved is thus the combination of mass and energy known as 'mass-energy'.
• In the context of mechanics, the law of conservation of mechanical energy sates that if the only force acting on an object is gravity then the sum of the kinetic and potential energies (the total mechanical energy) is constant.
• A machine is a device for multiplying force or changing the direction of force.
• Examples of simple machines include the lever, the pulley and the inclined plane.
• The mechanical advantage of a machine is defined as output force / input force. The m.a. can be greater than, less than or equal to 1.
• The efficiency of a machine is defined as useful work output / work input x 100%.
• The useful work output of a real machine is less than the total work input since friction converts some of the work input into useless heat. Therefore the efficiency of a real machine is always less than 100%.

### 6.10 Power

• Power is the rate at which work is done. Power = work done / time taken P = W / t
• Power is a scalar quantity, measured in joules per second or watts (W).