A vertical spring with force constant `k` is fixed on a table. A ball of mass `m` at a height `h` above the free upper end of the spring falls vertically on the spring, so that the spring is compressed by a distance `d`. The net work done in the process is

Options

A. `mg (h+d) + (1)/(2) kd^(2)`
B. `mg (h+d) - (1)/(2) kd^(2)`
C. `mg (h-d) - (1)/(2) kd^(2)`
D. `mg (h-d) + (1)/(2) kd^(2)`

Explanation

To solve the problem of finding the net work done when a ball of mass `m` falls from a height `h` onto a vertical spring with spring constant `k`, which compresses by a distance `d`, we can follow these steps:

### Step 1: Identify the initial potential energy
The initial potential energy (U_initial) of the ball when it is at height `h` above the spring is given by the gravitational potential energy formula:
\[ U_{\text{initial}} = mgh_{\text{initial}} = mg(h + d) \]
Here, `h + d` is the total height of the ball above the reference level (the top of the spring).

### Step 2: Identify the final potential energy
When the ball compresses the spring by a distance `d`, it comes to rest momentarily at maximum compression. At this point, the gravitational potential energy (U_final) is:
\[ U_{\text{final}} = 0 \]
because the ball is at the reference level (the top of the spring). The potential energy stored in the spring at maximum compression is:
\[ U_{\text{spring}} = \frac{1}{2} k d^2 \]

### Step 3: Calculate the change in potential energy
The change in potential energy (ΔU) as the ball moves from its initial position to the final position is:
\[ \Delta U = U_{\text{final}} - U_{\text{initial}} \]
Substituting the values we found:
\[ \Delta U = \left(0 - mg(h + d)\right) + \frac{1}{2} k d^2 \]
Thus, we can express the work done by conservative forces as:
\[ W_{\text{conservative}} = -\Delta U = mg(h + d) - \frac{1}{2} k d^2 \]

### Step 4: Write the final expression for the net work done
The net work done in the process is therefore:
\[ W_{\text{net}} = mg(h + d) - \frac{1}{2} k d^2 \]

### Final Answer
The net work done in the process is:
\[ W_{\text{net}} = mgh + mgd - \frac{1}{2} k d^2 \]

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