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Work
_1 Definition and Calculation_
Work done by a force is defined as the product of the force and
the displacement in the direction of the force.
Work done is a scalar and hence it can take on negative values just like temperature.
There are 3 basic or reference cases:
Case 1
object

force F

displacement d
Fig 1
Work done = F d
Force and displacement in
the same direction
(F and d are magnitudes)
Case 2

force F

displacement d
Fig 2
Force and displacement in
opposite direction
Work done = - F d
Case 3

force F

displacement d
Fig 3
Force and displacement
perpendicular to one another
Work done = 0
1
Any other scenario can be understood in terms of the 3 reference cases.
For example:

F
Fig 4


F
Fig 5
F sin
F cos

F sin

d


d
F cos
Work done = (F cos ) d
Work done = -(F cos) d
For the above cases, resolve the force into components parallel and perpendicular to the
displacement. Using the reference cases, the perpendicular component forces contribute
zero work done. Hence the work done is due only to the parallel components, giving the
results above.
Physicists like to deal with a single general law rather than many specific cases. In line with
that, a general formula that works for all cases is put together:


Work done = F d cos
where  is the angle between F and d
F is the magnitude of force
d is the magnitude of displacement
Note:
The angle  ranges from 0 to 1800.


 is the angle formed when both F and d are arranged with their tails together as
shown:
Fig 6

d

F

Fig 7

F

Fig 8


d

Fig 9

F


d

F



d

Can you see that when  = 0, the general formula reduces to case 1 formula F d?
Can you see that when  = 1800, the general formula reduces to case 2 formula - F d?
Can you see that when  = 900, the general formula reduces to case 3?
Can you see that in Fig 5, cos = - cos?
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_2 Meaning_
What does it mean by positive or negative work done on a system?
Doing positive work means energy is being transferred to the system while negative work
means energy is being removed from the system.
Hence, work done is actually a way of energy transfer, just like heating and cooling are
ways to give and remove energy from a system.
Consider Fig. 1 of Case 1. If the force is the only force acting and the body is at rest initially,
then the force is a net force that causes acceleration and a gain in speed. The kinetic
energy gained can be shown to be equal to the work done. Try proving it using kinematics
equations.
Consider another situation:
F
Force applied F and
sliding friction FR are
equal in magnitude
FR
d
Here, whatever energy given to the system via work done by F is immediately removed by
the negative work done by FR, leading to constant velocity.
Consider two masses attracted to each other by gravitational force and tension in a spring
respectively:
F
F
In each case, a force is applied to the mass on the left to hold it at a fixed position. At the
same time, another force is applied to pull the mass on the right away in a slow and steady
way until the final resting position. In both cases, the work done to pull apart the masses do
not lead to any increase in kinetic energy. Instead, the work done has become stored
gravitational and elastic potential energy respectively.
Tips on finding Work Done
1. Always be clear what is the system on which work is being done.
2. Always be clear which is the relevant force (out of many in a given scenario).
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