A block hangs freely from the end of a spring. A boy then slowly pushes the block upwards so that the spring becomes strain free. The gain in gravitational potential energy of the block during this process is not equal to

Options

A. (a) The work done by the boy against the gravitational force acting on the block
B. (b) The loss of energy stored in the spring minus the work done by the tension in the spring
C. (c) The work done on the block by the boy plus the loss of energy stored in the spring
D. (d) The work done on the block by the boy minus the work done by the tension in the spring plus the loss of energy stored in the spring
(e) The work done on the block by the boy minus the work done by the tension in the spring

Explanation

To solve the problem step by step, we need to analyze the situation where a block is hanging from a spring and is pushed upwards until the spring is strain-free. We want to determine which of the given options does not equal the gain in gravitational potential energy of the block during this process.

### Step 1: Understand the System
- A block of mass \( m \) is hanging from a spring. The weight of the block is \( mg \), where \( g \) is the acceleration due to gravity.
- When the block is hanging freely, it stretches the spring by an amount \( x \). The spring force is given by Hooke's Law as \( F_s = kx \), where \( k \) is the spring constant.

### Step 2: Determine the Relationship Between Forces
- At equilibrium (when the block is hanging freely), the gravitational force equals the spring force:
\[ mg = kx \]
From this, we can express \( x \):
\[ x = \frac{mg}{k} \]

### Step 3: Calculate the Gain in Gravitational Potential Energy
- The gain in gravitational potential energy (\( \Delta U \)) when the block is pushed upwards by the distance \( x \) is:
\[ \Delta U = mgx \]
- Substituting \( x \) from the previous step:
\[ \Delta U = mg \left(\frac{mg}{k}\right) = \frac{m^2g^2}{k} \]

### Step 4: Analyze Each Option
Now we will analyze each option to see which one does not equal the gain in gravitational potential energy.

**Option A**: Work done by the boy against the gravitational force acting on the block.
- This is equal to the gain in potential energy, so it's equal to \( \Delta U \).

**Option B**: Loss of energy stored in the spring minus the work done by the tension in the spring.
- The loss of energy stored in the spring is \( \frac{1}{2}kx^2 \) and the work done by the tension is also \( \frac{1}{2}kx^2 \). Thus, this results in \( 0 \), which is not equal to \( \Delta U \).

**Option C**: Work done on the block by the coil plus loss of energy stored in the spring.
- This results in \( \Delta U \), so it is equal.

**Option D**: Work done on the block by the boy minus work done by the tension in the spring plus the loss of energy stored in the spring.
- This also results in \( \Delta U \), so it is equal.

**Option E**: Work done on the block by the boy minus the work done by the tension in the spring.
- This results in \( \Delta U \), so it is equal.

### Conclusion
The only option that does not equal the gain in gravitational potential energy is **Option B**.