In which of the following cases no work is done by the force?

Options

A. (a) A man carrying a bucket of water, walking on a level road with a uniform velocity.
B. (b) A drop of rain falling vertically with a constant velocity.
C. (c) A man whirling a stone tied to a string in a circle with a cosntant speed.
D. (d) A man walking upon staircase

Explanation

To determine in which cases no work is done by the force, we can use the formula for work done:

\[ W = F \cdot s \cdot \cos(\theta) \]

Where:
- \( W \) is the work done,
- \( F \) is the force applied,
- \( s \) is the displacement,
- \( \theta \) is the angle between the force and the displacement.

Work is done when the force has a component in the direction of the displacement. If either the force is zero, the displacement is zero, or the angle \( \theta \) is 90 degrees (where \( \cos(90^\circ) = 0 \)), then no work is done.

Now, let's analyze each option step by step:

### Step 1: Analyze Option A
**Option A:** A man carrying a bucket of water walking on a level road with uniform speed.

- The man is moving at a constant speed, which implies that there is no net force acting on him (Newton's first law).
- Even though he is carrying the bucket, the force exerted by the man is vertical (upward) while the displacement is horizontal.
- The angle \( \theta \) between the force (upward) and the displacement (horizontal) is 90 degrees.

**Conclusion:** No work is done in this case.

### Step 2: Analyze Option B
**Option B:** A drop of rain falling vertically with constant velocity.

- The drop of rain is falling with constant velocity, which means there is no net force acting on it (again, due to Newton's first law).
- The force acting on the rain drop (gravity) is downward, and since it is falling straight down, the displacement is also downward.
- However, since it is at constant velocity, there is no net work being done on the drop.

**Conclusion:** No work is done in this case.

### Step 3: Analyze Option C
**Option C:** A man whirling a stone tied to a string in a circle with constant speed.

- The man is applying a force towards the center of the circle (centripetal force).
- The displacement of the stone is tangential to the circle, while the force is directed radially inward.
- The angle \( \theta \) between the force (inward) and the displacement (tangential) is 90 degrees.

**Conclusion:** No work is done in this case.

### Step 4: Analyze Option D
**Option D:** A man walking upon a staircase.

- In this case, the man is moving upward against gravity, which means he is doing work.
- The displacement is vertical, and the force exerted by the man is also vertical (against gravity).
- Since there is a change in height, work is done.

**Conclusion:** Work is done in this case.

### Final Answer:
No work is done in options A, B, and C.