A block of mass 2kg is lying on a rough inclined plane. The force needed to move the block up the plane with uniform velocity by applying a force parallel to the plane is 100N. The force needed to move the block up with an accele-ration of `2 ms^(-2)` is

Options

A. 100 N
B. 200 N
C. 96 N
D. 104 N

Answer

To solve the problem step by step, we need to analyze the forces acting on the block on the inclined plane.

### Step 1: Understanding the Forces
The block of mass \( m = 2 \, \text{kg} \) is on a rough inclined plane. The forces acting on the block include:
- The gravitational force \( mg \) acting downwards.
- The normal force \( N \) acting perpendicular to the inclined plane.
- The frictional force \( f \) acting down the plane (since the block is on a rough surface).
- The applied force \( F \) acting parallel to the inclined plane.

### Step 2: Forces when moving with uniform velocity
When the block is moving with uniform velocity, it is in equilibrium in the direction parallel to the incline. The force needed to move the block up the plane with uniform velocity is given as \( F = 100 \, \text{N} \). This means:
\[ F - mg \sin \theta - f = 0 \]
From this, we can conclude that:
\[ mg \sin \theta + f = 100 \, \text{N} \]

### Step 3: Forces when moving with acceleration
Now, when the block is moving up the incline with an acceleration \( a = 2 \, \text{m/s}^2 \), we can use Newton's second law:
\[ F - mg \sin \theta - f = ma \]
Substituting \( ma \) with \( 2m \) (since \( m = 2 \, \text{kg} \)):
\[ F - mg \sin \theta - f = 2m \]
\[ F - mg \sin \theta - f = 2 \times 2 \]
\[ F - mg \sin \theta - f = 4 \]

### Step 4: Relating the two scenarios
From the first scenario, we have:
\[ mg \sin \theta + f = 100 \, \text{N} \]
We can express \( f \) as:
\[ f = 100 - mg \sin \theta \]

Now, substituting this expression for \( f \) into the second equation:
\[ F - mg \sin \theta - (100 - mg \sin \theta) = 4 \]
This simplifies to:
\[ F - 100 = 4 \]
Thus, we find:
\[ F = 104 \, \text{N} \]

### Conclusion
The force needed to move the block up the inclined plane with an acceleration of \( 2 \, \text{m/s}^2 \) is \( 104 \, \text{N} \).