Heavier particle hits ground, lighter continues upward - inclined plane involved

7 \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-10_336_803_258_671} Two particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and 4 kg respectively are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(A\) is on a rough fixed slope which is at an angle of \(30 ^ { \circ }\) to the horizontal ground. Particle \(B\) hangs vertically below the pulley and is 0.5 m above the ground (see diagram). The coefficient of friction between the slope and particle \(A\) is 0.2 .

1. In the case where the system is in equilibrium with particle \(A\) on the point of moving directly up the slope, show that \(m = 5.94\), correct to 3 significant figures.
2. In the case where \(m = 3\), the system is released from rest with the string taut. Find the total distance travelled by \(A\) before coming to instantaneous rest. You may assume that \(A\) does not reach the pulley.

8. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a block \(A\) of mass 3 kg . Block \(A\) is held at rest on a smooth fixed plane. The plane is inclined at \(40 ^ { \circ }\) to the horizontal ground. The string lies along a line of greatest slope of the plane and passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a block \(B\) of mass 5 kg . Block \(B\) hangs freely at rest below the pulley, as shown in Figure 4. The system is released from rest with the string taut. By modelling the two blocks as particles,

1. find the tension in the string as \(B\) descends. After falling for 1.5 s , block \(B\) hits the ground and is immediately brought to rest. In its subsequent motion, \(A\) does not reach the pulley.
2. Find the speed of \(B\) at the instant it hits the ground.
3. Find the total distance moved up the plane by \(A\) before it comes to instantaneous rest. \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-28_97_141_2519_1804} \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-28_125_161_2624_1779}

7. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a small smooth fixed pulley at the top of the plane. The particle \(Q\) hangs freely below the pulley and 0.6 m above the ground, as shown in Figure 3. The part of the string from \(P\) to the pulley is parallel to a line of greatest slope of the plane. The system is released from rest with the string taut. For the motion before \(Q\) hits the ground,

1. 1. show that the acceleration of \(Q\) is \(\frac { 2 g } { 5 }\),
   2. find the tension in the string. On hitting the ground \(Q\) is immediately brought to rest by the impact.
2. Find the speed of \(P\) at the instant when \(Q\) hits the ground. In its subsequent motion \(P\) does not reach the pulley.
3. Find the total distance moved up the plane by \(P\) before it comes to instantaneous rest.
4. Find the length of time between \(Q\) hitting the ground and \(P\) first coming to instantaneous rest.

6 A block \(B\) of mass 0.85 kg lies on a smooth slope inclined at \(30 ^ { \circ }\) to the horizontal. \(B\) is attached to one end of a light inextensible string which is parallel to the slope. At the top of the slope, the string passes over a smooth pulley. The other end of the string hangs vertically and is attached to a particle \(P\) of mass 0.55 kg . The string is taut at the instant when \(P\) is projected vertically downwards.

1. Calculate
   1. the acceleration of \(B\) and the tension in the string,
   2. the magnitude of the force exerted by the string on the pulley. The initial speed of \(P\) is \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and after moving \(1.5 \mathrm {~m} P\) reaches the ground, where it remains at rest. \(B\) continues to move up the slope and does not reach the pulley.
   3. Calculate the total distance \(B\) moves up the slope before coming instantaneously to rest.

\includegraphics{figure_7} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a smooth plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). \(P\) lies on the plane and \(Q\) hangs vertically below the pulley at a height of 0.8 m above the floor (see diagram). The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. \(P\) is released from rest and \(Q\) moves vertically downwards.

1. Find the tension in the string and the magnitude of the acceleration of the particles. [5]

\(Q\) hits the floor and does not bounce. It is given that \(P\) does not reach the pulley in the subsequent motion.

1. Find the time, from the instant at which \(P\) is released, for \(Q\) to reach the floor. [2]
2. When \(Q\) hits the floor the string becomes slack. Find the time, from the instant at which \(P\) is released, for the string to become taut again. [4]

\includegraphics{figure_4} Figure 4 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small smooth light pulley \(A\) fixed at the top of the plane. The part of the string from \(P\) to \(A\) is parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below \(A\). The system is released from rest with the string taut.

1. Write down an equation of motion for \(P\) and an equation of motion for \(Q\). [4]
2. Hence show that the acceleration of \(Q\) is 0.98 m s\(^{-2}\). [2]
3. Find the tension in the string. [2]
4. State where in your calculations you have used the information that the string is inextensible. [1]

On release, \(Q\) is at a height of 0.8 m above the ground. When \(Q\) reaches the ground, it is brought to rest immediately by the impact with the ground and does not rebound. The initial distance of \(P\) from \(A\) is such that in the subsequent motion \(P\) does not reach \(A\). Find

1. the speed of \(Q\) as it reaches the ground, [2]
2. the time between the instant when \(Q\) reaches the ground and the instant when the string becomes taut again. [5]

\includegraphics{figure_3} Figure 3 shows two particles \(A\) and \(B\), of mass \(m\) kg and 0.4 kg respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely below \(P\). The system is released from rest with the string taut and \(B\) descends with acceleration \(\frac{1}{8}g\).

1. Write down an equation of motion for \(B\). [2]
2. Find the tension in the string. [2]
3. Prove that \(m = \frac{16}{35}\). [4]
4. State where in the calculations you have used the information that \(P\) is a light smooth pulley. [1]

On release, \(B\) is at a height of one metre above the ground and \(AP = 1.4\) m. The particle \(B\) strikes the ground and does not rebound.

1. Calculate the speed of \(B\) as it reaches the ground. [2]
2. Show that \(A\) comes to rest as it reaches \(P\). [5]

END

\includegraphics{figure_6} Two particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at \(30°\) to the horizontal. \(P\) has mass \(0.2\text{ kg}\) and is held at rest on the plane. \(Q\) has mass \(0.2\text{ kg}\) and hangs freely. The string is taut (see diagram). The coefficient of friction between \(P\) and the plane is \(0.4\). The particle \(P\) is released.

1. State the tension in the string before \(P\) is released, and find the tension in the string after \(P\) is released. [6]

\(Q\) strikes the floor and remains at rest. \(P\) continues to move up the plane for a further distance of \(0.8\text{ m}\) before it comes to rest. \(P\) does not reach the pulley.

1. Find the speed of the particles immediately before \(Q\) strikes the floor. [5]
2. Calculate the magnitude of the contact force exerted on \(P\) by the plane while \(P\) is in motion. [3]