Which one of the following forces cannot be expressed in terms of potential energy? Gravitational force Restoring force Frictional force Coulomb's force

 

❓ Question

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

• Gravitational force

• Restoring force

• Frictional force

• Coulomb's force

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🖼️ Question Image

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✍️ Short Solution

To solve this, we must identify which forces are conservative.

🔹 Step 1: Recall definitions

• A conservative force is one where the work done depends only on the initial and final positions, not on the path taken.

• For a conservative force, we can define a potential energy (U) such that

$$\vec{F}=-\mathrm{\nabla }U$$

• A non-conservative force depends on the path, so potential energy cannot be defined for it.

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🔹 Step 2: Check each force

1. Gravitational force ($F = mg$)

   • Conservative: work depends only on vertical displacement.

   • Potential energy: $U = m g h$ ✅

2. Restoring force (Hooke’s law: $F = -kx$)

   • Conservative: work depends only on displacement from equilibrium.

   • Potential energy: $U = \frac{1}{2} k x^2$ ✅

3. Frictional force

   • Non-conservative: work done depends on path length, not just displacement.

   • Cannot define a potential energy. ❌

4. Coulomb’s force ($F = k \frac{q_1 q_2}{r^2}$)

   • Conservative: work depends only on positions of charges.

   • Potential energy: $U = k \frac{q_1 q_2}{r}$ ✅

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🧮 Image Solution

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✅ Conclusion & Video Solution

✅ Answer:

$$\boxed{\text{Frictional force}}$$

📘 Concept Recap:

• Conservative forces → potential energy can be defined.

• Non-conservative forces → path-dependent work → no potential energy.

• Friction always removes energy as heat, depends on path, so no U can be assigned.

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