A small cart with a sphere suspended from ceiling by a string is moving up an inclined plane at a constant speed V. The direction of string supporting the sphere is `:`

Options

A. (a)vertical
B. (b)horizontal
C. (c)perpendicular to the inclined plane
D. (d)none of these

Answer

To solve the problem of determining the direction of the string supporting the sphere when a small cart is moving up an inclined plane at a constant speed \( V \), we can follow these steps:

### Step 1: Understand the scenario
We have a cart moving up an inclined plane with a sphere suspended from the ceiling by a string. The cart is moving at a constant speed \( V \).

**Hint:** Visualize the setup: draw the inclined plane, the cart, and the sphere hanging from the string.

### Step 2: Identify forces acting on the sphere
The forces acting on the sphere are:
- The gravitational force \( mg \) acting downward (where \( m \) is the mass of the sphere and \( g \) is the acceleration due to gravity).
- The tension \( T \) in the string acting upward along the direction of the string.

**Hint:** Remember that the gravitational force always acts downward towards the center of the Earth.

### Step 3: Analyze the motion of the cart and sphere
Since the cart is moving at a constant speed, there is no net acceleration acting on the sphere. This means that the forces acting on the sphere must be in equilibrium.

**Hint:** Use the concept of equilibrium: when an object is in equilibrium, the sum of the forces acting on it is zero.

### Step 4: Determine the direction of the tension
In equilibrium, the tension in the string must balance the gravitational force acting on the sphere. Since the cart is moving at a constant speed up the incline, the tension in the string will have a vertical component that balances the weight of the sphere.

**Hint:** Consider that if the cart were accelerating, the tension would have a different direction due to pseudo forces, but since it is constant speed, it remains vertical.

### Step 5: Conclude the direction of the string
Given that the sphere is in equilibrium and the cart is moving at a constant speed, the tension in the string must act vertically upward. Therefore, the direction of the string supporting the sphere is vertical.

**Hint:** Visualize the string: if the cart were to accelerate, the string would tilt, but since it is constant speed, the string remains vertical.

### Final Answer
The direction of the string supporting the sphere is vertical.