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Work and Work-Energy Theorem
Work
Work is defined as the scalar product of Force and Displacement. Work is done when a non-zero
net force acts on a moving object. If a force F moves an object over a distance d, work can be
found by using the formula:
G
F
θ
G
F
θ
G
d
G G
W = F ⋅ Δd
G
G
If F and d are parallel, the work is W =| F | ⋅ | Δd | . If they are not parallel, then the work is
W = |F|*|d|cos θ .
From this formula, one can conclude that for the work to be done, the following conditions must
be met:
The object must move (i.e. Δd ≠ 0). A force can be exerted on an object with no work done, e.g.
pushing on a wall, holding up 100 lb, etc.
If F and d are perpendicular, no work is done (cos θ = 0). For example, carrying something is not
work because the angle between the force (upwards) and displacement (horizontal) is equal to 0.
The SI unit of work is Joule (J), and 1 J = 1 N* 1 m.
If the angle between force and displacement is less than 900, cos θ >0, and the work is positive. If
90o < θ < 180o, cos θ <0, and the work is negative. Friction always does negative work because it
always acts in the direction opposite (180o) to the motion of the object.
Work either changes the velocity of an object or counteracts the work done by the opposite force.
For example, if a car is moving at a constant speed, the engine does work on the car to
counteract the work done by friction in the opposite direction. If an object is moving with constant
velocity, the net force acting on it is zero. In this case, the net amount of work done on it must be
zero, too (Wnet=Fnetd).
Energy
Energy is one of the most fundamental concepts of Physics. It is a very abstract concept that
does not have a single definition. Energy is a characteristic of a system. Energy in a system gives
it the capability to perform some operation. When work is done on a system or by a system, the
energy of the system changes. In other words, the system gains or loses energy through the
process of work. If positive work is done, the system gains energy. If work is negative, the system
loses energy. If no work is done, the total energy of the system does not change.
There are many types of energy. Kinetic energy (K or Ek) is the energy of motion. It depends only
upon the square of the speed and the mass of the object.
mV 2
K=
2
Work - Energy Theorem
An object is moving with an acceleration a over a distance d. Then according to the formula,
v 2f = vi2 + 2ad :
F
v 2f = vi2 + 2( )d
m
.
The work done on the object W is given by
m
1
1
W = Fd = (v 2f − vi2 ) = mv 2f − mv i2 = K f − K i
2
2
2
vi
G
F
vf
G
F
G
d
Or, we can rewrite it as
W = K f − Ki = ΔK
The work-energy theorem states that the total work done on the object by the forces equals the
change in kinetic energy.