# Work-mechanical energy principle

The work-kinetic energy theorem states that the net work or the work done by the net force is equal to the change in kinetic energy.

W_{net }= KE_{t} – KE_{o }= 1⁄2 m(v_{t}^{2} – v_{o}^{2})

W_{net} = There are two types of forces, namely conservative force, and non-conservative force. Thus, net work can be considered to be comprised of the work done by a conservative force and the work done by a non-conservative force.

W_{c} + W_{nc} = ΔKE

The work done by a conservative force is equal to the negative change in potential energy:

W_{c}= -ΔPE

– ΔPE + W_{nc} = ΔKE

W_{nc} = ΔPE + ΔKE

Wnc = ΔME

The equation above states that the work done by a non-conservative force on an object is equal to the change in the mechanical energy of the object. Mechanical energy = potential energy + kinetic energy. Potential energy can take the form of gravitational potential energy or elastic potential energy.

Example question: The work-mechanical energy theorem

A 2 kg box initially moves at a speed of 10 m/s. Shortly after, the box stops. The coefficient of kinetic friction between the box and the floor is 0.2. The gravitational acceleration is 10 m/s^{2}. How much is the box’s displacement?

Discussion:

Identified: m = 2 kg, v_{o} = 10 m/s, vt = 0, k = 0.2, w = m g = (1 kg)(10 m/s^{2}) = 10 kg m/s^{2} = 10 Newton,

Asked: the amount of the box’s displacement (s)

The work-mechanical energy theorem:

W_{nc} = ΔME

W_{nc} = ΔPE + ΔKE

The height (h) remains constant or there is no change in the height, so there is no change in the gravitational potential energy.

Wnc = ΔKE

The work done by the kinetic frictional force is:

W_{nc }= – f_{k} s = μ_{k} N -s = μ_{k} w -s = μk m g -s

W_{nc} = – (0.2)(2)(10)(s) = – (4)(s)

The kinetic frictional force does negative work (the kinetic frictional force is in opposite direction from the object’s displacement)

Change in the kinetic energy:

ΔKE = 1⁄2 m (v_{t}^{2} – v_{o}^{2}) = 1⁄2 (2)(0^{2 }– 10^{2}) = (0 – 100) = – 100

Object’s displacement:

W_{nc} = ΔKE

– (4)(s) = – 100

s = – 100 / – 4 = 25 meters