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.

20 conceptual questions and answers about the work done by conservative forces and potential energy:

**1. Question:** What are conservative forces? **Answer:** Conservative forces are forces for which the work done is independent of the path taken.

**2. Question:** How does the work done by a conservative force relate to the change in potential energy? **Answer:** The work done by a conservative force is equal to the negative change in potential energy.

**3. Question:** Name a common example of a conservative force. **Answer:** Gravitational force is a common example of a conservative force.

**4. Question:** Why is friction not considered a conservative force? **Answer:** Because the work done by friction depends on the path taken.

**5. Question:** How is gravitational potential energy of an object near the Earth’s surface expressed? **Answer:** Gravitational potential energy is expressed as $PE=mx g xh$, where m is mass, g is the acceleration due to gravity, and h is the height above a reference point.

**6. Question:** Can potential energy ever be negative? **Answer:** Yes, potential energy can be negative depending on the choice of the reference point.

**8. Question:** If the potential energy of a system increases, what happens to the kinetic energy, assuming no external work is done? **Answer:** The kinetic energy decreases by the same amount that potential energy increases, maintaining conservation of mechanical energy.

**9. Question:** What is the condition for the conservation of mechanical energy? **Answer:** Mechanical energy is conserved when only conservative forces act on a system.

**10. Question:** In a closed system where only conservative forces are acting, what can be said about the total mechanical energy? **Answer:** The total mechanical energy remains constant.

**11. Question:** How is the potential energy of an electric charge in an electric field determined? **Answer:** The potential energy $U=qxV$, where q is the charge and V is the potential.

**12. Question:** Why does raising an object in a gravitational field increase its potential energy? **Answer:** Because work is done against the gravitational force to raise the object, which gets stored as potential energy.

**13. Question:** How is the work done by a conservative force in a closed loop? **Answer:** The net work done by a conservative force over a closed loop is zero.

**14. Question:** How does potential energy relate to stability? **Answer:** At points where potential energy is a minimum, the system is stable.

**15. Question:** Can two objects at different heights have the same gravitational potential energy? **Answer:** Yes, they can, depending on the choice of the reference level for potential energy.

**16. Question:** Why do we often choose the ground as a reference point for gravitational potential energy? **Answer:** It provides a convenient and commonly understood reference, though any point can be chosen.

**17. Question:** In an isolated system with only conservative forces, if potential energy decreases by 10 J, how much does kinetic energy change? **Answer:** Kinetic energy increases by 10 J.

**18. Question:** How does potential energy relate to force in conservative fields? **Answer:** The force is the negative gradient of the potential energy.

**19. Question:** If an object is moved in the direction of a conservative force, how does its potential energy change? **Answer:** Its potential energy decreases.

**20. Question:** Is it possible for an object to have kinetic energy but zero potential energy? **Answer:** Yes, it depends on the choice of the reference point for potential energy.

These questions and answers cover fundamental concepts about conservative forces and potential energy, which form an integral part of classical mechanics.