WORK AND ENERGY

1. Definition of work: force applied in direction of displacement displacement. This
means that if the force F acts at an angle with respect to the direction of motion, then
W F d Fd
θ
×
= ⋅ =
􀁇 􀁇
cosθ
2 2-2
2. a) Work is a scalar - it has magnitude but no direction.
b) Work has dimensions: ( )
c) Work has units: 1 Newton 1 Metre 1 Joule (J)
3. Suppose you lift a mass of
M× LT− ×L =MLT
× ≡
20 kg through a distance of 2 metres. Then the work you
do is 20 kg 9.8 Newtons 2 metres 39.2 Joules. On the other hand, the force
of gravity is directed opposite to the force you exert
× × =
and the work done by gravity
is - 39.2 Joules.
4. What if the force varies with distance (say, a spring pulls harder as it becomes
longer). In that case, we should break up the distance over which the force acts into
small pieces so that the force is approximately constant over each bit. As we make the
pieces smaller and smaller, we will approach the exact result:
θ
F 􀁇
d

F
x →
(a,ka)
0
Δx = a / 4
1 2
3
4
a
to . This quantity is the work done by a force, constant or non-constant. So
if the force is known as a function of position, we can always find the work done by
calculating the defini
i f x x
te integral.
5. Just to check what our result looks like for a constant force, let us calculate W if F=b,
0
1( ) 1( ) 1( ) 1( )
4 4 4 4
a
a b + a b + a b + a b =ab ∴∫Fdx=ab
6. Now for a less trivial case: suppose that F=kx, i.e. the force increases linearly with x.
2 2
0
Area of shaded region 1 ( )( )
2 2 2
= a ka =ka ∴ ∫aFdx=ka
7. Energy is the capacity of a physical system to do work:
􀂃 it comes in many forms – mechanical, electrical, chemical, nuclear, etc
􀂃 it can be stored
􀂃 it can be converted into different forms
􀂃 it can never be created or destroyed
8. Accepting the fact that energy is conserved, let us derive an expression for the kinetic
energy of a body. Suppose a constant force accelerates a mass m from speed 0 to speed
v over a distance d. What is the work done by the force? Obviously the answer is:
2
. But and v2 2 . This gives ( ) v 1 v2. So, we
2 2
conclude that the work done by the force has gone into creating kinetic energy. and that
the amount of kinetic energy possessed by a
W Fd F ma ad W mad m d m
d
= = = = = =
1 2
2 body moving with speed v is mv .
9. The work done by a force is just the force multiplied by the distance – it does not
depend upon time. But suppose that the same amount of work is done in half the time.
F = b
F
0 Δx = a / 4 a x
1 2 3 4
We then say that the power is twice as much. We define:
Power = Work done
Time taken
1
2
If the force does not depend on time: Work v. Therefore, Power = v.
Time t
10.Let's work out an example. A constant force accelerates a bus (mass m) from speed v
to speed v over a dist
F xF F Δ
= =
Δ
ance d. What work is done by the engine?
2 2
2 1 2 1 2
2 2
2 1
2 1 1
Recall that for constant acceleration, v v 2 ( ) where: v = final velocity,
x = final position, v = initial velocity, x = initial position. Hence, v v . Now
2
calculate
a x x
a
d
− = −
−
=
2 2
2 1 2 2
2 1
2
the work done: = v v 1 v 1 v . So the
2 2 2
work done has resulted in an increase in the quantity 1 v , which is kinetic energy.
2
W Fd mad m d m m
d
m
−
= = = −