Projectile motion after leaving circle

7 A hollow cylinder of radius \(a\) is fixed with its axis horizontal. A particle \(P\), of mass \(m\), moves in part of a vertical circle of radius \(a\) and centre \(O\) on the smooth inner surface of the cylinder. The speed of \(P\) when it is at the lowest point \(A\) of its motion is \(\sqrt { \frac { 7 } { 2 } \mathrm { ga } }\). The particle \(P\) loses contact with the surface of the cylinder when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\theta = 60 ^ { \circ }\).
2. Show that in its subsequent motion \(P\) strikes the cylinder at the point \(A\).
   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 The training rig for a parachutist comprises a fixed platform and a fixed hook, \(H\). The platform is 3.5 m above horizontal ground level. The hook, which is not directly above the platform, is 6.5 m above the ground. One end of a light inextensible cord of length 4.5 m is attached to \(H\) and the other is attached to a trainee parachutist of mass 90 kg standing on the edge of the platform with the cord straight and taut. The trainee is then projected off the platform with a velocity of \(7 \mathrm {~ms} ^ { - 1 }\) perpendicular to the cord in a downward direction. The motion of the trainee all takes place in a single vertical plane and while the cord is attached to \(H\) it remains straight and taut. When the speed of the trainee reaches \(5.5 \mathrm {~ms} ^ { - 1 }\) the cord is detached from \(H\) and the trainee then moves under the influence of gravity alone until landing on the ground (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-6_615_1211_934_242} The trainee is modelled as a particle and air resistance is modelled as being negligible.

1. Show that at the instant before the cord is detached from \(H\), the tension in the cord has a magnitude of 1005.5 N . The point on the ground vertically below the edge of the platform is denoted by \(O\). The point on the ground where the trainee lands is denoted by \(T\).
2. Determine the distance \(O T\). The ground around \(T\) is in fact an elastic mat of thickness 0.5 m which is angled so that it is perpendicular to the direction of motion of the trainee on landing. The mat, which is very rough, is modelled as an elastic spring of natural length 0.5 m . It is assumed that the trainee strikes the mat at ground level and is brought to rest once the mat has been compressed by 0.3 m .
3. Determine the modulus of elasticity of the mat. Give your answer to the nearest integer.

4 A light inextensible taut rope, of length 4 m , is attached at one end \(A\) to the centre of the horizontal ceiling of a gym. The other end of the rope \(B\) is being held by a child of mass 35 kg . Initially the child is held at rest with the rope making an angle of \(60 ^ { \circ }\) to the downward vertical and it may be assumed that the child can be modelled as a particle attached to the end of the rope. The child is released at a height 5 m above the horizontal ground.

1. Show that the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the child when the rope makes an angle \(\theta\) with the downward vertical is given by \(v ^ { 2 } = 4 g ( 2 \cos \theta - 1 )\).
2. At the instant when \(\theta = 0 ^ { \circ }\), the child lets go of the rope and moves freely under the influence of gravity only. Determine the speed and direction of the child at the moment that the child reaches the ground.
3. The child returns to the initial position and is released again from rest. Find the value of \(\theta\) when the tension in the rope is three times greater than the tension in the rope at the instant the child is released.

7. \begin{figure}[h] \end{figure} Figure 3 shows a vertical cross-section through part of a ski slope consisting of a horizontal section \(A B\) followed by a downhill section \(B C\). The point \(O\) is on the same horizontal level as \(C\) and \(B C\) is a circular arc of radius 30 m and centre \(O\), such that \(\angle B O C = 90 ^ { \circ }\). A skier of mass 60 kg is skiing at \(12 \mathrm {~ms} ^ { - 1 }\) along \(A B\).

1. Assuming that friction and air resistance may be neglected, find the magnitude of the loss in reaction between the skier and the surface at \(B\).
   (4 marks)
   The skier subsequently leaves the slope at the point \(P\).
2. Find, correct to 3 significant figures, the speed at which the skier leaves the slope.
3. Find, correct to 3 significant figures, the speed of the skier immediately before hitting the ground again at the point \(D\) which is on the same horizontal level as \(C\).

7 A smooth track \(A B\) is in the shape of an arc of a circle with centre \(O\) and radius 1.4 m . The track is fixed in a vertical plane with \(A\) above the level of \(B\) and a point \(C\) on the track vertically below \(O\). Angle \(A O C\) is \(60 ^ { \circ }\) and angle \(C O B\) is \(30 ^ { \circ }\). Point \(C\) is 2.5 m vertically above the point \(F\), which lies in a horizontal plane. A particle of mass 0.4 kg is placed at \(A\) and projected down the track with an initial velocity of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle first hits the plane at point \(H\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-5_767_1265_488_415}

1. Find the magnitude of the contact force between the particle and the track when the particle is at \(B\). [5]
2. Find the distance \(F H\).

5 One end of a light inextensible string of length 0.8 m is attached to a fixed point, \(O\). The other end is attached to a particle \(P\) of mass \(1.2 \mathrm {~kg} . P\) hangs in equilibrium at a distance of 1.5 m above a horizontal plane. The point on the plane directly below \(O\) is \(F\).
\(P\) is projected horizontally with speed \(3.5 \mathrm {~ms} ^ { - 1 }\). The string breaks when \(O P\) makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downwards vertical through \(O\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-3_776_910_1242_244}

1. Find the magnitude of the tension in the string at the instant before the string breaks.
2. Find the distance between \(F\) and the point where \(P\) first hits the plane.

11 A particle \(P\) of mass 1 kg is fixed to one end of a light inextensible string of length 0.5 m . The other end of the string is attached to a fixed point O , which is 1.75 m above a horizontal plane. P is held with the string horizontal and taut. P is then projected vertically downwards with a speed of \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the tangential acceleration of P when OP makes an angle of \(20 ^ { \circ }\) with the horizontal. The string breaks when the tension in it is 32 N . At this point the angle between OP and the horizontal is \(\theta\).
2. Show that \(\theta = 23.1 ^ { \circ }\), correct to \(\mathbf { 1 }\) decimal place. Particle P subsequently hits the plane at a point A .
3. Determine the speed of P when it arrives at A .
4. Show that A is almost vertically below O .