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. 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.
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| 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. |
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.
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| 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.
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| The dot product of any two vectors is itself a scalar quantity.
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| Vectors can also be multiplied together to give a 'cross product'
which is itself a vector but this will not be explained here.
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6.4 Kinetic and potential energies
| Energy is the ability to do work.
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| Kinetic energy (motion energy) = ½ mass x speed2
K = ½ mv2
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| Potential energy is stored energy, often related to the
object's position
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| Gravitational potential energy = mass x acceleration of free fall x
height U = mgh
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| 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)
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| To avoid ambiguity, it is acceptable to distinguish between
gravitational potential energy and elastic potential energy by using
subscripts: Ugrav, Uel
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| The spring constant k is the ratio of the extending force to the
extension k = F/x
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| The sum of the kinetic energy and potential energy is 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.
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| 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'.
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| 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.
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| A machine is a device for
multiplying force or changing the direction of force.
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| Examples of simple machines include the lever, the pulley and the inclined plane.
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| 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.
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| The efficiency of a machine is defined as useful work output / work
input x 100%.
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| 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
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| Power is a scalar quantity, measured in joules per second or watts
(W). |
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