A man of mass `m` slides down along a rop which is connected to the celling of an elevator with deceleration `a` relative to the rop.If the elevator is going upward with an acceleration relative to the ground, then tension in the rop is

Options

A. `mg`
B. `m (g + 2a)`
C. `m (g + a)`
D. zero

Explanation

To solve the problem, we need to find the tension in the rope when a man of mass \( m \) slides down the rope in an elevator that is accelerating upwards with a certain acceleration. Here are the steps to derive the solution:

### Step 1: Understand the accelerations involved
- Let the acceleration of the elevator relative to the ground be \( a_e \).
- The man is sliding down with a deceleration \( a \) relative to the rope. This means that the man is accelerating downwards relative to the rope.

### Step 2: Determine the acceleration of the man relative to the ground
- The acceleration of the man relative to the rope is \( -a \) (since he is sliding down).
- The acceleration of the rope (elevator) is \( a_e \) (upwards).
- Therefore, the acceleration of the man relative to the ground can be expressed as:
\[ a_m = a_e - a \]

### Step 3: Set up the equation for tension in the rope
- According to Newton's second law, the net force acting on the man is equal to the mass times the acceleration of the man.
- The forces acting on the man are the tension \( T \) in the rope acting upwards and the weight \( mg \) acting downwards.
- The equation can be set up as:
\[ T - mg = m a_m \]
Substituting \( a_m \) from Step 2, we have:
\[ T - mg = m(a_e - a) \]

### Step 4: Solve for tension \( T \)
- Rearranging the equation gives:
\[ T = mg + m(a_e - a) \]
- Factor out \( m \):
\[ T = m(g + a_e - a) \]

### Final Answer
The tension in the rope is:
\[ T = m(g + a_e - a) \]