A spring, placed horizontally on a rough surface is compressed by a block of mass m, placed on the same surface so as to store maximum energy in the spring. If the coefficient of friction between the block and the surface is `mu`, the potential energy stored in the spring is

Options

A. `(mu^(2)m^(2)g^(2))/(2k)`
B. `(2mu m^(2)g^(2))/(k)`
C. `(mu^(2)m^(2)g)/(2k)`
D. `(3mu^(2)mg^(2))/(k)`

Answer

To solve the problem of determining the potential energy stored in a spring compressed by a block on a rough surface, we can follow these steps:

### Step 1: Understand the Forces Involved
When the block of mass \( m \) compresses the spring, it experiences two main forces: the spring force \( F_s \) acting to restore the spring to its equilibrium position and the frictional force \( F_f \) opposing the motion of the block. The maximum static friction force can be expressed as:
\[ F_f = \mu N \]
where \( \mu \) is the coefficient of friction and \( N \) is the normal force. For a horizontal surface, the normal force \( N \) is equal to the weight of the block, which is \( mg \). Thus, we have:
\[ F_f = \mu mg \]

### Step 2: Relate Spring Force to Friction
The spring force can be expressed using Hooke's Law as:
\[ F_s = kx \]
where \( k \) is the spring constant and \( x \) is the compression of the spring. For the block to remain at rest while compressing the spring, the spring force must equal the maximum friction force:
\[ kx = \mu mg \]

### Step 3: Solve for Compression \( x \)
From the equation \( kx = \mu mg \), we can solve for \( x \):
\[ x = \frac{\mu mg}{k} \]

### Step 4: Calculate the Potential Energy Stored in the Spring
The potential energy \( U \) stored in a compressed spring is given by the formula:
\[ U = \frac{1}{2} k x^2 \]
Substituting the expression for \( x \) we found in Step 3:
\[ U = \frac{1}{2} k \left( \frac{\mu mg}{k} \right)^2 \]
Simplifying this expression:
\[ U = \frac{1}{2} k \cdot \frac{\mu^2 m^2 g^2}{k^2} = \frac{\mu^2 m^2 g^2}{2k} \]

### Final Answer
Thus, the potential energy stored in the spring when compressed by the block is:
\[ U = \frac{\mu^2 m^2 g^2}{2k} \]

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