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Work, Energy and Power

When a force displaces an object by acting upon it, it is taken that work is done on the system. Force should be connected to displacement causally for work to take place. Daily life offers some good examples of work being done on a system - a horse pulling a cart through the road, a person lifting weight in the gym, a hospital staff pushing the mop across the floor, a batter hitting the baseball etc. In every situation described, force exerted by an agent is doing work on a system.


Calculating the Amount of Work Done by Forces

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Work done by a force on an object can be expressed mathematically as a product of force, displacement and cosine of the angle the force makes with the direction of displacement of the object. The expression reads as following

$W$ = $Fd\ cos\ (\theta)$

where F is the force, W is work done by the force, d is the displacement of the object and theta is the angle between the force vector and the displacement vector.

It follows directly from the equation that when net displacement due to a force is zero, the force does no work. The equation also implies that when a system moves along the direction of the force exerted, work is positive and when the system moves opposite to the direction of the force, work is taken to be negative. Work is expressed in the units of Joules (J) or kiloJoules (kJ).

Amount of Work Done by Forces

Potential Energy

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Potential energy is the energy stored in an object by virtue of its position or configuration. In other words, this energy has the potential to do work. Potential energy is expressed in the units of Joules(J) or kiloJoules(kJ). Next we discuss a few types of potential energy encountered in everyday life.
Gravitational Potential Energy:

When you stand at the top of a building you have more potential energy than when you are on the ground looking at it. This is because the earth attracts you with the gravitational force, doing work in the process. Gravitational potential energy is the energy of an object due to its height. The energy results from the gravitational pull of the Earth on the object. Gravitational potential energy is expressed as

$PE_{gravitational}$ = $mgh$

where m is the mass of the object, h is its height from the reference point (often taken as the ground) and g is the acceleration due to gravity.
Elastic Potential Energy:

Elastic potential energy is the energy stored in objects as the result of deformation. A good example is a spring, which comes back to its equilibrium configuration when stretched or compressed, doing work in the process. Elastic potential energy is expressed as

$PE_{elastic}$ = $\frac{1}{2}$ $kx^2$

where k is the spring constant and x is its displacement from the equilibrium position.
Electric Potential Energy:

When two opposite charges are held apart in a configuration they have more potential energy than when they are close together. If you remove the constraint, they will smash onto each other, thereby doing work.
Magnetic Potential Energy:

Similarly two magnets can attract one another and the larger the distance they are kept apart the higher the potential energy is of the configuration. If they are released, they will move towards and eventually stick to one another. The magnetic potential energy is then spent doing work.

Kinetic Energy

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Kinetic energy is the energy of an object due to its motion. All moving objects possess kinetic energy. The unit of kinetic energy is Joules (J) and kiloJoules (kJ). Just as we divide motion into two principal types, kinetic energy can be translational or rotational based on the type of motion involved.
Translational Kinetic Energy:

While moving along a curve, the kinetic energy of an object can be expressed as follows 

$KE$ = $\frac{1}{2}$ $mv^2$

where m is the mass of the object and v is the speed of the object.
Rotational Kinetic Energy:

While undergoing rotation, an extended object will have the following kinetic energy

$KE$ = $\frac{1}{2}$ $Iw^2$

where I represents the moment of inertia for the object in question, w being its angular speed.

Mechanical Energy

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Mechanical energy is the total energy a system has due to its configuration or position and motion. Therefore, mechanical energy of an object can be expressed as

$Mechanical\ Energy$ = $PE\ +\ KE$

Like potential energy and kinetic energy, mechanical energy also has the unit of Joules (J) and kiloJoules (kJ).


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Power is the rate of doing work for a system. In other words, power is the rate at which energy is being spent. Power is expressed as
$P$ = $\frac{W}{t}$

where P is the power, W is work done and t is the time spent doing the work. 

Power has SI unit of Watt (W) which is Joules per second.
$[W]$ = $\frac{[J]}{[s]}$

Work, Energy and Power Examples

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A block is residing on a surface. Find work done on the block in the following situations.

a) When the block is pushed to the right by a force of 8 N it is displaced from its original position by 10 m. 

b) When the block is pulled with a string that makes an angle of 60* with the horizontal it moves a distance of 32 m. Tension in the string is 10 N.

c) When the block is pushed downwards with a force of 50 N.


a) Work done on the block is W = Fd cos (theta)
    = (8 N)(10 m)(cos 0*)
    = 80 J
b) Work done on the block is W= F.d cos (theta)
    =(10 N)(32 m)(cos 60*)
    =160 J
c) Normally, a downward force will not be able to displace a block. Therefore,
    Work done on the block is W = Fd cos (theta)
    = (50 N)(0 m)(cos 90*)
    = 0
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Work and Power Work and Kinetic Energy
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