# Work done by Conservative Forces, Potential Energy

Observe an object which moves vertically upwards and then return to its initial position after reaching a maximum height. When the object is moving vertically upwards, weight does negative work on the object. When the object is moving upwards, the object’s height increases. Therefore, the object’s gravitational potential energy increases as well. It can be concluded that the negative work done by weight is equal to the increase in the object’s gravitational potential energy (PE).

W = – m g h

W = – m g (h_{2} – h_{1}) = – (PE_{2 }– PE_{1}) = -ΔPE

When the object is moving vertically downwards, gravitational force does positive work on the object. When the object is moving vertically downwards, the object’s height decreases. Therefore, the object’s gravitational potential energy decreases as well. It can be concluded that the positive work done by the gravitational force on the object is equal to the decrease in the object’s gravitational potential energy.

W = m g h

W = m g (h_{1 }– h_{2}) = m g h_{1 }– m g h_{2} = – (m g h_{2} – m g h_{1}) = – (PE_{2 }– PE_{1}) = -ΔPE

Observe an object which is pressed to the left along with the end of a spring (see figure 3). When the object is moving to the left, the spring force does negative work on the object. When the object is moving to the left, the spring deviation increases. Therefore, the spring potential energy increases as well. It can be concluded that the negative work done by the spring is equal to the increase in the spring potential energy. When the object is moving to the right, the spring force does positive work on the object. When the object is moving to the right, the spring deviation decreases. Therefore, the spring potential energy decreases as well. It can be concluded that the positive work done by the spring on the object is equal to the decrease in the spring potential energy.

According to the explanation above, it can be said that the work done by a conservative force is equal to the change in the object’s potential energy. When a conservative force does positive work, the potential energy decreases. Conversely, when a conservative force does negative work, the potential energy increases. Hence, the work done by a conservative force is equal to the negative change in potential energy.

W_{c} = -ΔPE

Example question 6: Work by a conservative force and potential energy

An object with a mass of 1 kg is at the height of 5 meters above the ground level. The gravitational acceleration is 10 m/s^{2}. Determine (a) the work done by weight when the object is displaced to a height of 10 meters above the ground level, (b) the work required to displace the object to a height of 10 meters, and (c) the change in the object’s gravitational potential energy when the object is moving to a height of 10 meters.

Discussion:

Identified: m = 1 kg, g = 10 m/s^{2} ,

(a) Work by weight

W = – m g Δh

= – (1 kg)(10 m/s^{2})(5 m) = – 50 Joule

The weight does negative work because the direction of the weight is opposite to the direction of the object’s displacement.

(weight is in down direction, object’s displacement is in up direction).

(b) Work by lift force

So that the object can be lifted, the minimum lift force must be equal to the weight.

= (1 kg)(10 m/s^{2})(5 m) = 50 Joule

W = w h = m g Δh

The work done by the lift force is positive as the force direction is the same as the displacement.

(force lift is in up direction, displacement is in up direction)

(c) Change in potential energy

ΔPE = m g Δh

ΔPE = 50 Joule

The object’s gravitational potential energy increases by 50 Joule.