Question

Which one of the following forces cannot be expressed in terms of potential energy?

• A. Coulomb's force
• B. Gravitational force
• C. Frictional force
• D. Restoring force

Hint:

A key characteristic of conservative forces is that the work they do is path-independent, and they allow for the definition of a potential energy. Non-conservative forces, like friction, result in energy dissipation, and their work depends on the path taken.

Answer 1

The Correct Option is C

To determine which force cannot be expressed in terms of potential energy, we need to consider the characteristics of each listed force:

1. Coulomb's Force: This is the electrostatic force between two charged particles. It can be expressed in terms of potential energy because it is a conservative force. The potential energy \( U \) associated with Coulomb's force can be written as \(U = \frac{k \cdot q_1 \cdot q_2}{r}\) where \( k \) is Coulomb’s constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.
2. Gravitational Force: This force is exerted by masses due to gravity. It is also a conservative force and can be expressed in terms of potential energy, given by \(U = -\frac{G \cdot m_1 \cdot m_2}{r}\) where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between their centers.
3. Restoring Force: This is the force that brings a system back to equilibrium, such as in springs (Hooke's law). It is conservative, and its potential energy is expressed as \(U = \frac{1}{2} k x^2\) where \( k \) is the spring constant and \( x \) is the displacement.
4. Frictional Force: This force opposes motion and is non-conservative. Because of its dissipative nature (it converts mechanical energy into heat), it cannot be expressed as potential energy. Instead, it is derived from the velocity of the moving object and acts to restrict motion, making it non-reversible and energy-dissipative.

Based on this analysis, frictional force is the force that cannot be expressed in terms of potential energy.

Conclusion: The frictional force is a non-conservative force and is the correct answer.

Answer 2

Step 1: Understand the concept of conservative forces and potential energy.
A force is said to be conservative if the work done by the force in moving a particle between two points is independent of the path taken. For a conservative force, it is possible to define a potential energy function \( U \) such that the force \( \vec{F} \) is related to the potential energy by \( \vec{F} = -\nabla U \) (in three dimensions) or \( F = -\frac{dU}{dx} \) (in one dimension). Equivalently, the work done by a conservative force over a closed path is zero.
Step 2: Analyze each of the given forces.
(1) Coulomb's force:
The electrostatic force between two charges is a conservative force. The work done by the Coulomb's force depends only on the initial and final positions of the charges, not on the path taken. The potential energy associated with the Coulomb's force is the electrostatic potential energy.
(2) Gravitational force:
The gravitational force between two masses is also a conservative force. The work done by the gravitational force depends only on the initial and final positions of the masses, and the potential energy associated with it is the gravitational potential energy.
(3) Frictional force:
Frictional force is a non-conservative force. The work done by friction depends on the path taken. For example, the work done against friction is greater along a longer path between two points. Also, the work done by friction over a closed path is not zero; it is always negative (dissipative).
Therefore, frictional force cannot be expressed in terms of a potential energy function. The energy dissipated by friction is converted into heat.
(4) Restoring force (e.g., spring force):
The restoring force exerted by a spring is a conservative force. The work done by the spring force depends only on the initial and final extensions or compressions of the spring, and the potential energy associated with it is the elastic potential energy.
Step 3: Identify the force that cannot be expressed in terms of potential energy.
Based on the analysis above, frictional force is a non-conservative force and cannot be expressed in terms of potential energy.