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_t² – v_o²)

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.

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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/s². 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/s²) = 10 kg m/s² = 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)

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Change in the kinetic energy:

ΔKE = 1⁄2 m (v_t² – v_o²) = 1⁄2 (2)(0² – 10²) = (0 – 100) = – 100

Object’s displacement:

W_nc = ΔKE

– (4)(s) = – 100

s = – 100 / – 4 = 25 meters