In order for an object to gain energy, work must be done on it by an external force. When work is done on an object by a force acting parallel to its displacement the formula is:

 

work_done = force x displacement

 

For positive work to be done, F and s must be parallel and pointed in the same direction. The unit used to measure work and energy is a joule. [J = kg m²/sec² = Nm]

 

 

If we look at the forces on an object being pulled across a table's surface there would be three: F, the applied force, N, the normal or supporting force supplied by the table, and mg, its weight or the gravitational force of attraction to the earth.

 

 

The normal force and the object's weight are in static equilibrium (they are balanced forces), the applied force, F, is an unbalanced force and will result in the object being accelerated across the top of the table's surface in the same direction as the force. This acceleration will change the object's velocity and subsequently its kinetic energy. We say that this applied force is doing work on the object. The amount of work done by F is directly proportional to the distance through which the force is applied as it pulls the object across the table's surface.

 

net Work_done = (net F)s

 

 

By using Newton's second law, net F = ma, our equation becomes

net Work_done = (ma)s
net Work_done = m(as)

 

Remembering the kinematics equation v_f² = v_o² + 2as and solving for "as" let's our equation become

net Work_done = m[½(v_f² - v_o²)]
net Work_done = ½(mv_f² - mv_o²)
net Work_done = ½mv_f² - ½mv_o²
net W_done = ΔKE

 

The relationship we just derived is called the energy-work theorem..

 

This statement tells us that when an external force does work on an object it will change the object's kinetic energy; that is, it will cause the object to either gain or lose speed. When more than one force is acting on an object, all forces that are either parallel or antiparallel to the direction the object moves will do work. If the object's velocity remains constant, that just means that the work done by opposing forces (for example, a forward applied force, F, and an opposing force, friction) are equal.

 

Note that if F and s are perpendicular to each other no work is done on the object.  In our  example of the block being dragged across the table, neither the normal force nor the weight would do any work on the block since they act at right angles to the direction of the block's motion. Another example would be when a satellite is being held in circular orbit by the force of gravity. Note that since the satellite's speed and orbital radius remain constant, no energy is being changed; therefore, no work is being done on the satellite.

 

 

Work done by conservative forces, or path-independent forces, results in changes in the object's potential energy.

 

 

 

When you observe an object falling, it loses potential energy (height) while it gains kinetic energy (speed). That is, in the absence of another external, non-conservative force, such as friction, pushing/pulling, or tensions in strings, the total amount of potential energy before the fall equals the total amount of kinetic energy after the fall and the energy-work theorem is restated as the Law of Conservation of Energy:

 

Work_done = ΔKE
Work_{done conservative force}= - ΔPE

- ΔPE = ΔKE
- (PE_f - PE_o) = KE_f - KE_o
- PE_f + PE_o = KE_f - KE_o

KE_o+ PE_o = KE_f + PE_f

 

For projectiles in freefall this statement of conservation of energy can be used to compare the energies at two different locations (A and B) in its trajectory:

 

PE_A + KE_A = PE_B + KE_B

 

Remember that energy (as well as work) is a scalar. The velocity in the KE formula represents the RESULTANT velocity or the actual instantaneous SPEED of the object. You do not take x and y components. This formula could also be used to determine the speed of a cart on a frictionless roller coaster, or a bead on a frictionless wire.

 

 

The formula used to calculate the power delivered or developed to complete a specified amount of work in a given time is

 

Power = work/time

 

Power is measured in watts. [watt = J/sec]

 

A second form of power is easily derived.

 

Power = work/time
Power = (Fs) / t
Power = F (s/t)
Power = Fv

 

This expression can only be used when either the velocity is constant or when only an instantaneous value for power is being requested.