Problem:
A block of mass 2 kg 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. Find the force needed to move the block up the plane with an acceleration of 2 m/s².

Solution:

Given:
Mass of the block (m) = 2 kg
Force required to move the block up the plane with uniform velocity (F₁) = 100 N
Acceleration (a) = 2 m/s²

Force required to move the block up the plane with uniform velocity (F₁):
When the block is moving up the plane with a uniform velocity, the force required can be found using the formula:

F₁ = m * g * sinθ + μ * m * g * cosθ

where,
m = mass of the block
g = acceleration due to gravity
θ = angle of inclination
μ = coefficient of friction between the block and the inclined plane

Since the block is at rest initially, the force required to move it up the plane with an acceleration of 2 m/s² can be found using Newton's second law of motion:

F = m * a

Force required to move the block up the plane with an acceleration of 2 m/s² (F):
Using Newton's second law of motion,

F = m * a
F = 2 kg * 2 m/s²
F = 4 N

Explanation:
To move the block up the plane with a uniform velocity, a force of 100 N is required. This force is the sum of the gravitational force component and the frictional force component. The gravitational force component can be calculated by multiplying the mass of the block by the acceleration due to gravity and the sine of the angle of inclination. The frictional force component can be calculated by multiplying the coefficient of friction between the block and the inclined plane, the mass of the block, the acceleration due to gravity, and the cosine of the angle of inclination.

To move the block up the plane with an acceleration of 2 m/s², a force of 4 N is required. This force is equal to the product of the mass of the block and the desired acceleration. According to Newton's second law of motion, the force required to produce an acceleration on an object is directly proportional to the mass of the object and the desired acceleration.

Therefore, to move the block up the plane with an acceleration of 2 m/s², a force of 4 N is required.