The potential energy of a weight less spring compressed by a distance a is proportional to

Options

A. a
B. `a^2`
C. `a^-2`
D. `a^0`

Answer

To find the potential energy of a weightless spring compressed by a distance \( a \), we can follow these steps:

### Step 1: Understand the Spring Force
The spring force \( F \) exerted by a spring is given by Hooke's Law:
\[ F = -k \cdot x \]
where \( k \) is the spring constant and \( x \) is the displacement from the natural length of the spring. In this case, when the spring is compressed by a distance \( a \), the force exerted by the spring is:
\[ F = -k \cdot a \]

### Step 2: Determine the External Force
To compress the spring, an external force \( F_{\text{external}} \) must be applied in the opposite direction of the spring force. Therefore, the external force required to compress the spring by a distance \( a \) is:
\[ F_{\text{external}} = k \cdot a \]

### Step 3: Calculate the Work Done
The potential energy \( U \) stored in the spring when it is compressed is equal to the work done by the external force in compressing the spring. The work done \( W \) by the external force when compressing the spring from its natural length (0) to a distance \( a \) can be calculated using the integral:
\[ W = \int_0^a F_{\text{external}} \, dx \]
Substituting \( F_{\text{external}} = kx \):
\[ W = \int_0^a kx \, dx \]

### Step 4: Perform the Integration
Now, we can perform the integration:
\[ W = k \int_0^a x \, dx = k \left[ \frac{x^2}{2} \right]_0^a = k \left( \frac{a^2}{2} - 0 \right) = \frac{1}{2} k a^2 \]

### Step 5: Conclusion
The potential energy \( U \) stored in the spring when compressed by a distance \( a \) is:
\[ U = \frac{1}{2} k a^2 \]
This shows that the potential energy is proportional to the square of the distance \( a \).

### Final Answer
The potential energy of a weightless spring compressed by a distance \( a \) is proportional to \( a^2 \).

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