Question

Explain the motion of a body on an inclined plane.

Answer 1

Motion on an Inclined Plane

Consider an inclined plane as shown in the figure that makes an angle 0 with the horizontal plane OA Assume an object of mass m moving on this plane. Initially, let us suppose that the object does not move due to friction. Various forces act on the object.
(i) Weight of the object, W = mg
(ii) Normal reaction, N
(iii) Frictional force, f_s
In the vertical direction, there is no acceleration. Balancing the components of weight, W = mg,

mg sinθ = f_s …………. (1)
and, mg cosθ = N …………. (2)
Now, if we move the plane OB upwards so that 0 increases, then at a position, the object will start moving on the inclined plane OR At this position
(f_s)_max = μ_sN
Where, μ_s is the coefficient of static friction between the object and the plane. In this position, let θ = θ_s (maximum value of the angle) and (f_s)_max are placed in the above expressions,
mg sinθ_s = μ_sN
mg cosθ_s = N
From above equations,

tanθ_s = μ_s ……….. (3)
From equation (3), we can determine the coefficient of static friction between the inclined plane and the surface of the object.
When the value of θ is slightly more than θ_s, then a small net force acts on the object and object starts to slide. We further increase the value of θ so that the object moves downward with accelerated motion.

Then, we keep on decreasing the value of θ so that the object moves with a constant velocity at θ = θ_k,. At this position, kinetic frictional force acts on the object.
Substituting f_k in place of f_s, μ_k in place of μ_s and θ_k in place of θ_s in equation (1 & 2), we get the following equations :
mg sinθ_k = f_k ………….. (4)

mg cosθ_k = N ………….. (5)
Again, putting f_k = μ_kN in above expressions,
tanθ_k = μ_k ……………. (6)
From equation (6), we can determine the coefficient of kinetic friction between the inclined plane and the surface of the object.
Knowing the distances of the horizontal plane OA and inclined plane OR we can find tanθ_s and tanθ_k using the relation, tanθ = \(\frac{A B}{O A}=\frac{\sqrt{O B^{2}-O A^{2}}}{O A}\) this way, we can find out the coefficient of friction.
Note that BA is perpendicular to OA.