Introduction:

When an object is placed on a smooth inclined plane, it experiences a gravitational force acting vertically downwards and a normal force acting perpendicular to the plane. The inclined plane provides a component of the normal force acting parallel to the plane, which opposes the gravitational force. To keep the object stationary relative to the incline, the horizontal acceleration imparted to the inclined plane must counterbalance this component of the normal force.

Understanding the situation:

- Inclined plane angle: The inclined plane has an angle of 1° with the horizontal.
- Forces acting on the object: The object experiences a gravitational force (mg) acting vertically downwards and a normal force (N) acting perpendicular to the inclined plane.
- Component of normal force: The component of the normal force acting parallel to the inclined plane is given by Nsinθ, where θ is the angle of inclination.
- Horizontal acceleration: The horizontal acceleration imparted to the inclined plane is denoted by a.

Resolving forces:

- Resolving the gravitational force: The gravitational force (mg) can be resolved into two components: mgcosθ perpendicular to the inclined plane and mgsinθ parallel to the inclined plane.
- Resolving the normal force: The normal force (N) can be resolved into two components: Ncosθ perpendicular to the inclined plane and Nsinθ parallel to the inclined plane.

Condition for object to be stationary:

For the object to be stationary relative to the incline, the horizontal acceleration (a) imparted to the inclined plane must counterbalance the component of the normal force (Nsinθ) acting parallel to the inclined plane. Mathematically, this can be expressed as:

a = Nsinθ

Calculating the normal force:

To calculate the normal force (N), we need to consider the forces acting perpendicular to the inclined plane. The gravitational force (mg) can be resolved into mgcosθ perpendicular to the inclined plane, and the normal force (Ncosθ) also acts in the same direction. Therefore, the net force acting perpendicular to the inclined plane is given by:

Net force = mgcosθ + Ncosθ

To keep the object stationary, the net force acting perpendicular to the inclined plane should be zero. Therefore, we can equate the net force to zero and solve for N:

mgcosθ + Ncosθ = 0

N = -mgcotθ

Substituting values and calculating the horizontal acceleration:

Now, substituting the value of the normal force (N = -mgcotθ) into the equation a = Nsinθ, we get:

a = (-mgcotθ)sinθ

Simplifying this expression, we have:

a = -mgcosθ

Therefore, the horizontal acceleration (a) imparted to the inclined plane should be equal to -mgcosθ to keep the object stationary relative to the incline. The negative sign indicates that the acceleration is in the opposite direction to the motion of the object.

Conclusion:

To summarize, in order for an object to be stationary relative to a smooth inclined plane with an angle of inclination of 1°, the horizontal acceleration imparted to the inclined plane should be equal to -mgcosθ. This acceleration counterbalances the component of the normal force acting parallel to the inclined plane, ensuring that the object remains stationary