Collision on slope

7 A bead, \(A\), of mass 0.1 kg is threaded on a long straight rigid wire which is inclined at \(\sin ^ { - 1 } \left( \frac { 7 } { 25 } \right)\) to the horizontal. \(A\) is released from rest and moves down the wire. The coefficient of friction between \(A\) and the wire is \(\mu\). When \(A\) has travelled 0.45 m down the wire, its speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(\mu = 0.25\).
   Another bead, \(B\), of mass 0.5 kg is also threaded on the wire. At the point where \(A\) has travelled 0.45 m down the wire, it hits \(B\) which is instantaneously at rest on the wire. \(A\) is brought to instantaneous rest in the collision. The coefficient of friction between \(B\) and the wire is 0.275 .
2. Find the time from when the collision occurs until \(A\) collides with \(B\) again.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-10_501_416_262_861} Particles \(P\) and \(Q\) have masses \(m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively. The particles are initially held at rest 6.4 m apart on the same line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.8\) (see diagram). Particle \(P\) is released from rest and slides down the line of greatest slope. Simultaneously, particle \(Q\) is projected up the same line of greatest slope at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between each particle and the plane is 0.6 .

1. Show that the acceleration of \(Q\) up the plane is \(- 11.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the time for which the particles are in motion before they collide.
3. The particles coalesce on impact. Find the speed of the combined particle immediately after the impact.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 \includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-12_560_716_258_712} Particles of masses 1.5 kg and 3 kg lie on a plane which is inclined at an angle of \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The section of the plane from \(A\) to \(B\) is smooth and the section of the plane from \(B\) to \(C\) is rough. The 1.5 kg particle is held at rest at \(A\) and the 3 kg particle is in limiting equilibrium at \(B\). The distance \(A B\) is \(x \mathrm {~m}\) and the distance \(B C\) is 4 m (see diagram).

1. Show that the coefficient of friction between the particle at \(B\) and the plane is 0.75 .
   The 1.5 kg particle is released from rest. In the subsequent motion the two particles collide and coalesce. The time taken for the combined particle to travel from \(B\) to \(C\) is 2 s . The coefficient of friction between the combined particle and the plane is still 0.75 .
2. Find \(x\).
3. Find the total loss of energy of the particles from the time the 1.5 kg particle is released until the combined particle reaches \(C\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

\includegraphics{figure_7} The diagram shows two particles \(P\) and \(Q\) which lie on a line of greatest slope of a plane \(ABC\). Particles \(P\) and \(Q\) are each of mass \(m\) kg. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.6\). The length of \(AB\) is 0.75 m and the length of \(BC\) is 3.25 m. The section \(AB\) of the plane is smooth and the section \(BC\) is rough. The coefficient of friction between each particle and the section \(BC\) is 0.25. Particle \(P\) is released from rest at \(A\). At the same instant, particle \(Q\) is released from rest at \(B\).

1. Verify that particle \(P\) reaches \(B\) 0.5 s after it is released, with speed \(3\) m s\(^{-1}\). [3]
2. Find the time that it takes from the instant the two particles are released until they collide. [4]

The two particles coalesce when they collide. The coefficient of friction between the combined particle and the plane is still 0.25.

1. Find the time that it takes from the instant the particles collide until the combined particle reaches \(C\). [5]

Fig. 1.1 shows block A of mass 2.5 kg which has been placed on a long, uniformly rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.8\). The coefficient of friction between A and the slope is 0.85. \includegraphics{figure_1}

1. Calculate the maximum possible frictional force between A and the slope. Show that A will remain at rest. [6]

With A still at rest, block B of mass 1.5 kg is projected down the slope, as shown in Fig. 1.2. B has a speed of 16 m s\(^{-1}\) when it collides with A. In this collision the coefficient of restitution is 0.4, the impulses are parallel to the slope and linear momentum parallel to the slope is conserved.

1. Show that the velocity of A immediately after the collision is 8.4 m s\(^{-1}\) down the slope. Find the velocity of B immediately after the collision. [6]
2. Calculate the impulse on B in the collision. [3]

The blocks do not collide again.

1. For what length of time after the collision does A slide before it comes to rest? [4]

\includegraphics{figure_12} A particle \(P\) of mass 2 kg can move along a line of greatest slope on a smooth plane, inclined at \(30°\) to the horizontal. \(P\) is initially at rest at a point on the plane, and a force of constant magnitude 20 N is applied to \(P\) parallel to and up the slope (see diagram).

1. Copy and complete the diagram, showing all forces acting on \(P\). [1]
2. Find the velocity of \(P\) in terms of time \(t\) seconds, whilst the force of 20 N is applied. [4]

After 3 seconds the force is removed at the instant that \(P\) collides with a particle of mass 1 kg moving down the slope with speed 5 m s\(^{-1}\). The coefficient of restitution between the particles is 0.2.

1. Express the velocity of \(P\) as a function of time after the collision. [6]