- #1

Sihas N

- 4

- 2

- Homework Statement
- The sphere A is released from the top of a rough inclined plane with an inclination of \mathbit{\alpha} to the horizontal. Let the length of the inclined plane be \mathbit{d}_\mathbf{1}. After a time interval ∆t1, another sphere B is released from the top with an initial velocity \mathbit{v}_\mathbf{1} and simultaneously a third sphere C is also projected vertically upwards the plane with an initial velocity \mathbit{v}_\mathbf{2} \left(\mathbit{v}_\mathbf{2}>\mathbit{v}_\mathbf{1}\right). Both spheres B and C collide with the sphere A at the same instant. The sphere B then rebounds instantaneously for a distance ∆d upwards the vertical plane and the sphere C immediately stops at the moment of collision. The inclination of the plane becomes \mathbit{\beta} at the instance of collision \left(\mathbit{\alpha}>\mathbit{\beta}\right). The time interval between the stoppage and the restart of the motion is ∆t2. Then all three spheres start to roll down the plane. All three spheres are made of different materials. The coefficient of friction between the plane and the sphere A, B and C is \mathbit{\mu}_\mathbf{1},\ \mathbit{\mu}_\mathbf{2} and \mathbit{\mu}_\mathbf{3} respectively. The coefficient of restitution between the sphere A and the sphere B is \mathbit{e}_\mathbf{1} and between the sphere A and sphere C is \mathbit{e}_\mathbf{2}. Then all three spheres reach a rough horizontal floor where the coefficient of friction between the surface and the sphere A, B and C is \mathbit{\mu}_\mathbf{4},\ \mathbit{\mu}_\mathbf{5} and \mathbit{\mu}_\mathbf{6} respectively. Spheres A and C come to rest keeping distances between each other as \mathbit{l} where the sphere C comes to rest at the base of another inclined plane, whereas the sphere B keeps moving with a uniform velocity which it attains after the collision. The distance between the first inclined plane and the second inclined plane is \mathbit{d}_\mathbf{2} Once they come to the horizontal floor, an impulsive force of I is applied on to the sphere B. It moves with an acceleration \mathbit{f} and collides again with the sphere A and stops. Thereafter, the sphere A too moves with a uniform velocity \mathbit{v}_\mathbf{3} and then collides again with the sphere C. As soon as the sphere C is collided, it is moved upwards the rough plane inclined at an angle \mathbit{\theta} to the horizontal where the coefficient of friction between the plane and the sphere is \mathbit{\mu}_\mathbf{7}. But the sphere C could only move upwards for a distance \mathbit{d}_\mathbf{3} of the total length of the inclined plane. Then it rolls back down and re-collides with the sphere A and it too moves backwards with a velocity \mathbit{v}_\mathbf{4} but stops just touching the sphere B, such that none of the spheres move thereafter. Assume that the sphere C stops immediately after colliding with the sphere A. This whole motion takes a time \mathbit{T}. Let the masses of the spheres A, B and C be \mathbit{m}_\mathbf{1},\ \mathbit{m}_\mathbf{2} and \mathbit{m}_\mathbf{3} respectively. Considering the motion from the first inclined plane to the other inclined plane as positive draw the necessary VT graph for the motion of three spheres until they all come to rest.

- Relevant Equations
- Since the question just asks to draw the necessary VT graph, I did not involve any equations.

I couldn't draw the motion after the collision, since the whole angular displacement of the plane got me confused.