6.05e Radial/tangential acceleration

6 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-4_497_524_276_804} A particle \(P\) of mass 0.4 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point \(O\). With the string taut the particle is travelling in a circular path in a vertical plane. The angle between the string and the downward vertical is \(\theta ^ { \circ }\) (see diagram). When \(\theta = 0\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. At the instant when the string is horizontal, find the speed of \(P\) and the tension in the string.
2. At the instant when the string becomes slack, find the value of \(\theta\).

3 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-3_387_181_274_982} \(A\) and \(B\) are fixed points with \(B\) at a distance of 1.8 m vertically below \(A\). One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to \(A\), and one end of an identical elastic string is attached to \(B\). A particle \(P\) of weight 12 N is attached to the other ends of the strings (see diagram).

1. Verify that \(P\) is in equilibrium when it is at a distance of 1.05 m vertically below \(A\). \(P\) is released from rest at the point 1.2 m vertically below \(A\) and begins to move.
2. Show that, when \(P\) is \(x \mathrm {~m}\) below its equilibrium position, the tensions in \(P A\) and \(P B\) are \(( 18 + 40 x ) \mathrm { N }\) and \(( 6 - 40 x ) \mathrm { N }\) respectively.
3. Show that \(P\) moves with simple harmonic motion of period 0.777 s , correct to 3 significant figures.
4. Find the speed with which \(P\) passes through the equilibrium position. \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-3_540_655_1564_744} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. With the string taut and horizontal, \(P\) is projected with a velocity of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downward. \(P\) begins to move in a vertical circle with centre \(O\). While the string remains taut the angular displacement of \(O P\) is \(\theta\) radians from its initial position, and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).

5 A light elastic string of natural length 1.6 m has modulus of elasticity 120 N . One end of the string is attached to a fixed point \(O\) and the other end is attached to a particle \(P\) of weight 1.5 N . The particle is released from rest at the point \(A\), which is 2.1 m vertically below \(O\). It comes instantaneously to rest at \(B\), which is vertically above \(O\).

1. Verify that the distance \(A B\) is 4 m .
2. Find the maximum speed of \(P\) during its upward motion from \(A\) to \(B\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_351_442_303_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_394_648_260_1018} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A light inextensible string of length \(0.8 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.4 kg and 0.58 kg respectively, attached to its ends. The string passes over a smooth horizontal cylinder of radius 0.8 m , which is fixed with its axis horizontal and passing through a fixed point \(O\). The string is held at rest in a vertical plane perpendicular to the axis of the cylinder, with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the cylinder through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).

2 A particle of mass 0.4 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.5 m . The particle is projected horizontally with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the point 0.5 m vertically below \(O\). The particle moves in a complete circle. Find the tension in the string when

1. the string is horizontal,
2. the particle is vertically above \(O\).

7 \includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-4_351_314_255_861} One end of a light elastic string, of natural length \(\frac { 2 } { 3 } R \mathrm {~m}\) and with modulus of elasticity 1.2 mgN , is attached to the highest point \(A\) of a smooth fixed sphere with centre \(O\) and radius \(R \mathrm {~m}\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string and is in contact with the surface of the sphere, where the angle \(A O P\) is equal to \(\theta\) radians (see diagram).

1. Given that \(P\) is in equilibrium at the point where \(\theta = \alpha\), show that \(1.8 \alpha - \sin \alpha - 1.2 = 0\). Hence show that \(\alpha = 1.18\) correct to 3 significant figures. \(P\) is now released from rest at the point of the surface of the sphere where \(\theta = \frac { 2 } { 3 }\), and starts to move downwards on the surface. For an instant when \(\theta = \alpha\),
2. state the direction of the acceleration of \(P\),
3. find the magnitude of the acceleration of \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-4_524_611_255_703} A hollow cylinder is fixed with its axis horizontal. The inner surface of the cylinder is smooth and has radius 0.6 m . A particle \(P\) of mass 0.45 kg is projected horizontally with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the lowest point of a vertical cross-section of the cylinder and moves in the plane of the cross-section, which is perpendicular to the axis of the cylinder. While \(P\) remains in contact with the surface, its speed is \(v \mathrm {~ms} ^ { - 1 }\) when \(O P\) makes an angle \(\theta\) with the downward vertical at \(O\), where \(O\) is the centre of the cross-section (see diagram). The force exerted on \(P\) by the surface is \(R \mathrm {~N}\).

1. Show that \(v ^ { 2 } = 4.24 + 11.76 \cos \theta\) and find an expression for \(R\) in terms of \(\theta\).
2. Find the speed of \(P\) at the instant when it leaves the surface.

5 \includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_449_442_1281_794} A fixed smooth sphere of radius 0.6 m has centre \(O\) and highest point \(T\). A particle of mass \(m \mathrm {~kg}\) is released from rest at a point \(A\) on the sphere, such that angle \(T O A\) is \(\frac { \pi } { 6 }\) radians. The particle leaves the surface of the sphere at \(B\) (see diagram).

1. Show that \(\cos T O B = \frac { \sqrt { 3 } } { 3 }\).
2. Find the speed of the particle at \(B\).
3. Find the transverse acceleration of the particle at \(B\).

6 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772} A hollow cylinder is fixed with its axis horizontal. \(O\) is the centre of a vertical cross-section of the cylinder and \(D\) is the highest point on the cross-section. \(A\) and \(C\) are points on the circumference of the cross-section such that \(A O\) and \(C O\) are both inclined at an angle of \(30 ^ { \circ }\) below the horizontal diameter through \(O\). The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle \(P\), of mass \(m \mathrm {~kg}\), and a particle \(Q\), of mass \(5 m \mathrm {~kg}\), are simultaneously released from rest from \(A\) and \(C\), respectively, inside the cylinder. \(P\) and \(Q\) collide; the coefficient of restitution between them is 0.95 .

1. Show that, immediately after the collision, \(P\) moves with speed \(6.3 \mathrm {~ms} ^ { - 1 }\), and find the speed and direction of motion of \(Q\).
2. Find, in terms of \(m\), an expression for the normal reaction acting on \(P\) when it subsequently passes through \(D\).

5 \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_576_535_258_804} A particle \(P\) of mass 0.3 kg is moving in a vertical circle. It is attached to the fixed point \(O\) at the centre of the circle by a light inextensible string of length 1.5 m . When the string makes an angle of \(40 ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Air resistance may be neglected.

1. Find the radial and transverse components of the acceleration of \(P\) at this instant. In the subsequent motion, with the string still taut and making an angle \(\theta ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Use conservation of energy to show that \(v ^ { 2 } \approx 19.7 + 29.4 \cos \theta ^ { \circ }\).
3. Find the tension in the string in terms of \(\theta\).
4. Find the value of \(v\) at the instant when the string becomes slack. \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_574_842_1640_664} A step-ladder is modelled as two uniform rods \(A B\) and \(A C\), freely jointed at \(A\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) in contact with a rough horizontal surface. The rods have equal lengths; \(A B\) has weight 150 N and \(A C\) has weight 270 N . The point \(A\) is 2.5 m vertically above the surface, and \(B C = 1.6 \mathrm {~m}\) (see diagram).
5. Find the horizontal and vertical components of the force acting on \(A C\) at \(A\).
6. The coefficient of friction has the same value \(\mu\) at \(B\) and at \(C\), and the step-ladder is on the point of slipping. Giving a reason, state whether the equilibrium is limiting at \(B\) or at \(C\), and find \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-4_648_227_269_982} Two points \(A\) and \(B\) lie on a vertical line with \(A\) at a distance 2.6 m above \(B\). A particle \(P\) of mass 10 kg is joined to \(A\) by an elastic string and to \(B\) by another elastic string (see diagram). Each string has natural length 0.8 m and modulus of elasticity 196 N . The strings are light and air resistance may be neglected.
7. Verify that \(P\) is in equilibrium when \(P\) is vertically below \(A\) and the length of the string \(P A\) is 1.5 m . The particle is set in motion along the line \(A B\) with both strings remaining taut. The displacement of \(P\) below the equilibrium position is denoted by \(x\) metres.
8. Show that the tension in the string \(P A\) is \(245 ( 0.7 + x )\) newtons, and the tension in the string \(P B\) is \(245 ( 0.3 - x )\) newtons.
9. Show that the motion of \(P\) is simple harmonic.
10. Given that the amplitude of the motion is 0.25 m , find the proportion of time for which \(P\) is above the mid-point of \(A B\).

2

1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e0e5580a-e1f0-46f8-9304-2a96533af186-03_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Express \(v ^ { 2 }\) in terms of \(\theta\).
   2. Show that \(R = 5.88 \sin \theta - 3.43\).
   3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
   4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.

2

1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-3_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Express \(v ^ { 2 }\) in terms of \(\theta\).
   2. Show that \(R = 5.88 \sin \theta - 3.43\).
   3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
   4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.

2

1. A light inextensible string has length 1.8 m . One end of the string is attached to a fixed point O , and the other end is attached to a particle of mass 5 kg . The particle moves in a complete vertical circle with centre O , so that the string remains taut throughout the motion. Air resistance may be neglected.
   1. Show that, at the highest point of the circle, the speed of the particle is at least \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   2. Find the least possible tension in the string when the particle is at the lowest point of the circle.
2. Fig. 2 shows a hollow cone mounted with its axis of symmetry vertical and its vertex V pointing downwards. The cone rotates about its axis with a constant angular speed of \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). A particle P of mass 0.02 kg is in contact with the rough inside surface of the cone, and does not slip. The particle P moves in a horizontal circle of radius 0.32 m . The angle between VP and the vertical is \(\theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b209dbe7-769c-4301-a2f3-108c27c8cefb-3_588_510_1046_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} In the case when \(\omega = 8.75\), there is no frictional force acting on P .
   1. Show that \(\tan \theta = 0.4\). Now consider the case when \(\omega\) takes a constant value greater than 8.75.
   2. Draw a diagram showing the forces acting on P .
   3. You are given that the coefficient of friction between P and the surface is 0.11 . Find the maximum possible value of \(\omega\) for which the particle does not slip.

2

1. A small ball of mass 0.01 kg is moving in a vertical circle of radius 0.55 m on the smooth inside surface of a fixed sphere also of radius 0.55 m . When the ball is at the highest point of the circle, the normal reaction between the surface and the ball is 0.1 N . Modelling the ball as a particle and neglecting air resistance, find
   1. the speed of the ball when it is at the highest point of the circle,
   2. the normal reaction between the surface and the ball when the vertical height of the ball above the lowest point of the circle is 0.15 m .
2. A small object Q of mass 0.8 kg moves in a circular path, with centre O and radius \(r\) metres, on a smooth horizontal surface. A light elastic string, with natural length 2 m and modulus of elasticity 160 N , has one end attached to Q and the other end attached to O . The object Q has a constant angular speed of \(\omega\) rad s \(^ { - 1 }\).
   1. Show that \(\omega ^ { 2 } = \frac { 100 ( r - 2 ) } { r }\) and deduce that \(\omega < 10\).
   2. Find expressions, in terms of \(r\) only, for the elastic energy stored in the string, and for the kinetic energy of Q . Show that the kinetic energy of Q is greater than the elastic energy stored in the string.
   3. Given that the angular speed of Q is \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tension in the string.

2

1. Fig. 2 shows a light inextensible string of length 3.3 m passing through a small smooth ring R of mass 0.27 kg . The ends of the string are attached to fixed points A and B , where A is vertically above \(B\). The ring \(R\) is moving with constant speed in a horizontal circle of radius \(1.2 \mathrm {~m} , \mathrm { AR } = 2.0 \mathrm {~m}\) and \(\mathrm { BR } = 1.3 \mathrm {~m}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-3_570_659_493_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Show that the tension in the string is 6.37 N .
   2. Find the speed of R .
2. One end of a light inextensible string of length 1.25 m is attached to a fixed point O . The other end is attached to a particle P of mass 0.2 kg . The particle P is moving in a vertical circle with centre O and radius 1.25 m , and when P is at the highest point of the circle there is no tension in the string.
   1. Show that when P is at the highest point its speed is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the instant when the string OP makes an angle of \(60 ^ { \circ }\) with the upward vertical, find
   2. the radial and tangential components of the acceleration of P ,
   3. the tension in the string.

3 A particle P of mass 0.6 kg is connected to a fixed point O by a light inextensible string of length 1.25 m . When it is 1.25 m vertically below \(\mathrm { O } , \mathrm { P }\) is set in motion with horizontal velocity \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then moves in part of a vertical circle with centre O and radius 1.25 m . When OP makes an angle \(\theta\) with the downward vertical, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 3.1. \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 11.5 + 24.5 \cos \theta\).
2. Find the tension in the string in terms of \(\theta\).
3. Find the speed of P at the instant when the string becomes slack. A second light inextensible string, of length 0.35 m , is attached to P , and the other end of this string is attached to a point C which is 1.2 m vertically below O . The particle P now moves in a horizontal circle with centre C and radius 0.35 m , as shown in Fig. 3.2. The speed of P is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-3_518_488_1701_826} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
4. Find the tension in the string OP and the tension in the string CP.

2

1. A particle P , of mass 48 kg , is moving in a horizontal circle of radius 8.4 m at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in contact with a smooth horizontal surface. A light inextensible rope of length 30 m connects P to a fixed point A which is vertically above the centre C of the circle, as shown in Fig. 2.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-3_526_490_482_870} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure}
   1. Given that \(V = 3.5\), find the tension in the rope and the normal reaction of the surface on P .
   2. Calculate the value of \(V\) for which the normal reaction is zero.
2. The particle P , of mass 48 kg , is now placed on the highest point of a fixed solid sphere with centre O and radius 2.5 m . The surface of the sphere is smooth. The particle P is given an initial horizontal velocity of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and it then moves in part of a vertical circle with centre O and radius 2.5 m . When OP makes an angle \(\theta\) with the upward vertical and P is still in contact with the surface of the sphere, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction of the sphere on P is \(R \mathrm {~N}\), as shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-3_590_617_1706_804} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
   1. Show that \(v ^ { 2 } = u ^ { 2 } + 49 - 49 \cos \theta\).
   2. Find an expression for \(R\) in terms of \(u\) and \(v\).
   3. Given that P loses contact with the surface of the sphere at the instant when its speed is \(4.15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(u\).

2 A light inextensible string of length 5 m has one end attached to a fixed point A and the other end attached to a particle P of mass 0.72 kg . At first, P is moving in a vertical circle with centre A and radius 5 m . When P is at the highest point of the circle it has speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the tension in the string when the speed of P is \(15 \mathrm {~ms} ^ { - 1 }\). The particle P now moves at constant speed in a horizontal circle with radius 1.4 m and centre at the point C which is 4.8 m vertically below A .
2. Find the tension in the string.
3. Find the time taken for P to make one complete revolution. Another light inextensible string, also of length 5 m , now has one end attached to P and the other end attached to the fixed point B which is 9.6 m vertically below A . The particle P then moves with constant speed \(7 \mathrm {~ms} ^ { - 1 }\) in the circle with centre C and radius 1.4 m , as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86d79489-aec1-4c94-bef6-45b007f818a0-3_693_465_1078_817} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
4. Find the tension in the string PA and the tension in the string PB .

2

1. A fixed solid sphere with a smooth surface has centre O and radius 0.8 m . A particle P is given a horizontal velocity of \(1.2 \mathrm {~ms} ^ { - 1 }\) at the highest point on the sphere, and it moves on the surface of the sphere in part of a vertical circle of radius 0.8 m .
   1. Find the radial and tangential components of the acceleration of P at the instant when OP makes an angle \(\frac { 1 } { 6 } \pi\) radians with the upward vertical. (You may assume that P is still in contact with the sphere.)
   2. Find the speed of P at the instant when it leaves the surface of the sphere.
2. Two fixed points R and S are 2.5 m apart with S vertically below R . A particle Q of mass 0.9 kg is connected to R and to S by two light inextensible strings; Q is moving in a horizontal circle at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with both strings taut. The radius of the circle is 2.4 m and the centre C of the circle is 0.7 m vertically below S, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3f674569-7e99-4ba8-84f1-a1eb438e30ed-2_547_720_1946_644} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Find the tension in the string RQ and the tension in the string \(S Q\).

2 A fixed hollow sphere with centre O has an inside radius of 2.7 m . A particle P of mass 0.4 kg moves on the smooth inside surface of the sphere. At first, P is moving in a horizontal circle with constant speed, and OP makes a constant angle of \(60 ^ { \circ }\) with the vertical (see Fig. 2.1). \begin{figure}[h] \end{figure}

1. Find the normal reaction acting on P .
2. Find the speed of P . The particle P is now placed at the lowest point of the sphere and is given an initial horizontal speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves in part of a vertical circle. When OP makes an angle \(\theta\) with the upward vertical and P is still in contact with the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\) (see Fig. 2.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-3_716_778_1653_696} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. Find \(v ^ { 2 }\) in terms of \(\theta\).
4. Show that \(R = 4.16 - 11.76 \cos \theta\).
5. Find the speed of P at the instant when it leaves the surface of the sphere.

2 A particle P of mass 0.3 kg is connected to a fixed point O by a light inextensible string of length 4.2 m . Firstly, P is moving in a horizontal circle as a conical pendulum, with the string making a constant angle with the vertical. The tension in the string is 3.92 N .

1. Find the angle which the string makes with the vertical.
2. Find the speed of P . P now moves in part of a vertical circle with centre O and radius 4.2 m . When the string makes an angle \(\theta\) with the downward vertical, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2). You are given that \(v = 8.4\) when \(\theta = 60 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2a4afead-e772-4d86-bc8d-86ffa5bca507-2_382_648_1985_751} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
3. Find the tension in the string when \(\theta = 60 ^ { \circ }\).
4. Show that \(v ^ { 2 } = 29.4 + 82.32 \cos \theta\).
5. Find \(\theta\) at the instant when the string becomes slack.

1 A fixed solid sphere has centre O and radius 2.6 m . A particle P of mass 0.65 kg moves on the smooth surface of the sphere. The particle P is set in motion with horizontal velocity \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the highest point of the sphere, and moves in part of a vertical circle. When OP makes an angle \(\theta\) with the upward vertical, and P is still in contact with the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 52.92 - 50.96 \cos \theta\).
2. Find, in terms of \(\theta\), the normal reaction acting on P .
3. Find the speed of P at the instant when it leaves the surface of the sphere. The particle P is now attached to one end of a light inextensible string, and the other end of the string is fixed to a point A , vertically above O , such that AP is tangential to the sphere, as shown in Fig. 1. P moves with constant speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with radius 2.4 m on the surface of the sphere. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3ec81c4e-e0fa-43d9-9c79-ef9df746be8f-2_1100_634_1089_753} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
4. Find the tension in the string and the normal reaction acting on P .

2 A hollow hemisphere has internal radius 2.5 m and is fixed with its rim horizontal and uppermost. The centre of the hemisphere is O . A small ball B of mass 0.4 kg moves in contact with the smooth inside surface of the hemisphere. At first, B is moving at constant speed in a horizontal circle with radius 1.5 m , as shown in Fig. 2.1. \begin{figure}[h] \end{figure}

1. Find the normal reaction of the hemisphere on \(B\).
2. Find the speed of \(\mathbf { B }\). The ball B is now released from rest on the inside surface at a point on the same horizontal level as O . It then moves in part of a vertical circle with centre O and radius 2.5 m , as shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-3_378_663_1427_740} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. Show that, when \(B\) is at its lowest point, the normal reaction is three times the weight of \(B\). For an instant when the normal reaction is twice the weight of \(\mathbf { B }\), find
4. the speed of \(\mathbf { B }\),
5. the tangential component of the acceleration of \(\mathbf { B }\).

2

1. A particle P of mass 0.2 kg is connected to a fixed point O by a light inextensible string of length 3.2 m , and is moving in a vertical circle with centre O and radius 3.2 m . Air resistance may be neglected. When P is at the highest point of the circle, the tension in the string is 0.6 N .
   1. Find the speed of P when it is at the highest point.
   2. For an instant when OP makes an angle of \(60 ^ { \circ }\) with the downward vertical, find
      (A) the radial and tangential components of the acceleration of P ,
      (B) the tension in the string.
2. A solid cone is fixed with its axis of symmetry vertical and its vertex V uppermost. The semivertical angle of the cone is \(36 ^ { \circ }\), and its surface is smooth. A particle Q of mass 0.2 kg is connected to V by a light inextensible string, and Q moves in a horizontal circle at constant speed, in contact with the surface of the cone, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-3_455_609_950_808} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The particle Q makes one complete revolution in 1.8 s , and the normal reaction of the cone on Q has magnitude 0.75 N .
   1. Find the tension in the string.
   2. Find the length of the string.

2

1. Fig. 2 shows a car of mass 800 kg moving at constant speed in a horizontal circle with centre C and radius 45 m , on a road which is banked at an angle of \(18 ^ { \circ }\) to the horizontal. The forces shown are the weight \(W\) of the car, the normal reaction, \(R\), of the road on the car and the frictional force \(F\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-3_286_970_402_561} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Given that the frictional force is zero, find the speed of the car.
   2. Given instead that the speed of the car is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the frictional force and the normal reaction.
2. One end of a light inextensible string is attached to a fixed point O , and the other end is attached to a particle P of mass \(m \mathrm {~kg}\). Starting with the string taut and P vertically below \(\mathrm { O } , \mathrm { P }\) is set in motion with a horizontal velocity of \(7 \mathrm {~ms} ^ { - 1 }\). It then moves in part of a vertical circle with centre O . The string becomes slack when the speed of P is \(2.8 \mathrm {~ms} ^ { - 1 }\). Find the length of the string. Find also the angle that OP makes with the upward vertical at the instant when the string becomes slack.

2 A particle P of mass 0.25 kg is connected to a fixed point O by a light inextensible string of length \(a\) metres, and is moving in a vertical circle with centre O and radius \(a\) metres. When P is vertically below O , its speed is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When OP makes an angle \(\theta\) with the downward vertical, and the string is still taut, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the tension in the string is \(T \mathrm {~N}\), as shown in Fig. 2. \begin{figure}[h] \end{figure}

1. Find an expression for \(v ^ { 2 }\) in terms of \(a\) and \(\theta\), and show that $$T = \frac { 17.64 } { a } + 7.35 \cos \theta - 4.9 .$$
2. Given that \(a = 0.9\), show that P moves in a complete circle. Find the maximum and minimum magnitudes of the tension in the string.
3. Find the largest value of \(a\) for which P moves in a complete circle.
4. Given that \(a = 1.6\), find the speed of P at the instant when the string first becomes slack.