Conical pendulum – horizontal circle in free space (no surface)

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A toy aircraft of mass 0.5 kg is attached to one end of a light inextensible string of length 9 m . The other end of the string is attached to a fixed point \(O\). The aircraft moves with constant speed in a horizontal circle. The string is taut, and makes an angle of \(60 ^ { \circ }\) with the upward vertical at \(O\) (see diagram). In a simplified model of the motion, the aircraft is treated as a particle and the force of the air on the aircraft is taken to act vertically upwards with magnitude 8 N . Find

1. the tension in the string,
2. the speed of the aircraft.

2 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-2_561_570_1274_790} A particle of mass 0.15 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. The particle moves with constant speed in a horizontal circle. The magnitude of the acceleration of the particle is \(7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The string makes an angle of \(\theta ^ { \circ }\) with the downward vertical, as shown in the diagram. Find

1. the value of \(\theta\) to the nearest whole number,
2. the tension in the string,
3. the speed of the particle.

3 \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-3_437_567_269_788} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) moves with constant speed in a horizontal circle, with the string taut and inclined at \(35 ^ { \circ }\) to the vertical. \(O P\) rotates with angular speed \(2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about the vertical axis through \(O\) (see diagram). Find

1. the value of \(L\),
2. the speed of \(P\) in \(\mathrm { m } \mathrm { s } ^ { - 1 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-3_456_511_260_817} A particle of mass 0.24 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. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T \mathrm {~N}\). The acceleration of the particle has magnitude \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(\tan \theta = 0.75\) and find the value of \(T\).
2. Find the speed of the particle.

2 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-2_565_549_438_797} A non-uniform rod \(A B\) of weight 6 N rests in limiting equilibrium with the end \(A\) in contact with a rough vertical wall. \(A B = 1.2 \mathrm {~m}\), the centre of mass of the rod is 0.8 m from \(A\), and the angle between \(A B\) and the downward vertical is \(\theta ^ { \circ }\). A force of magnitude 10 N acting at an angle of \(30 ^ { \circ }\) to the upwards vertical is applied to the rod at \(B\) (see diagram). The rod and the line of action of the 10 N force lie in a vertical plane perpendicular to the wall. Calculate

1. the value of \(\theta\),
2. the coefficient of friction between the rod and the wall.

5 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-3_365_679_264_733} A uniform metal frame \(O A B C\) is made from a semicircular \(\operatorname { arc } A B C\) of radius 1.8 m , and a straight \(\operatorname { rod } A O C\) with \(A O = O C = 1.8 \mathrm {~m}\) (see diagram).

1. Calculate the distance of the centre of mass of the frame from \(O\). A uniform semicircular lamina of radius 1.8 m has weight 27.5 N . A non-uniform object is formed by attaching the frame \(O A B C\) around the perimeter of the lamina. The object is freely suspended from a fixed point at \(A\) and hangs in equilibrium. The diameter \(A O C\) of the object makes an angle of \(22 ^ { \circ }\) with the vertical.
2. Calculate the weight of the frame.

3 A particle \(P\) of mass 0.5 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\); one force has magnitude \(\left( 1 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and acts in the positive \(x\)-direction, and the other force has magnitude \(8 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the negative \(x\)-direction.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 2 + 0.6 x ^ { 2 } - 16 \mathrm { e } ^ { - x }\).
2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~ms} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 3\).
3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_259_745_278_740} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A small sphere \(A\) of mass 0.15 kg is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(A\) moves with constant speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.35 m , and is continuously in contact with both the plane base and the curved surface of the cylinder. Fig. 1 shows a vertical cross-section of the cylinder through its axis. Find the magnitude of the force exerted on \(A\) by

1. \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-03_62_37_2659_1914}

1. \begin{figure}[h] \end{figure} A garden game is played with a small ball \(B\) of mass \(m\) attached to one end of a light inextensible string of length 13l. The other end of the string is fixed to a point \(A\) on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius \(5 l\) and centre \(C\), where \(C\) is vertically below \(A\). Modelling the ball as a particle, find

1. the tension in the string,
2. the speed of the ball.

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The particle moves in a horizontal circle with constant angular speed \(\sqrt { 58.8 } \mathrm { rad } \mathrm { s } ^ { - 1 }\). The centre \(O\) of the circle is vertically below \(A\) and the string makes a constant angle \(\theta ^ { \circ }\) with the downward vertical, as shown in Figure 2. Given that the tension in the string is 1.2 mg , find

1. the value of \(\theta\)
2. the length of the string.

2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point \(O\). The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of the circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-2_344_340_1418_842}

1. Show that the tension in the string is 22.6 N , correct to three significant figures.
2. Find the speed of the particle.

6 A light inextensible string has one end attached to a particle, \(P\), of mass 2 kg . The other end of the string is attached to the fixed point \(A\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed in a horizontal circle of radius 0.8 m and centre \(B\). The tension in the string is 34 N . The string is inclined at an angle \(\theta\) to the vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-4_760_816_1436_596}

1. Find the angle \(\theta\).
2. Find the speed of the particle.
3. Find the time taken for the particle to make one complete revolution.

2 A particle of mass 0.3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The particle moves in a horizontal circle of radius 0.343 m , with centre vertically below \(A\), at a constant angular speed of \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find the tension in the string and the angle at which the string is inclined to the vertical.

2

1. A particle P of mass 0.6 kg is connected to a fixed point by a light inextensible string of length 2.8 m . The particle P moves in a horizontal circle as a conical pendulum, with the string making a constant angle of \(55 ^ { \circ }\) with the vertical.
   1. Find the tension in the string.
   2. Find the speed of P .
2. A turntable has a rough horizontal surface, and it can rotate about a vertical axis through its centre O . While the turntable is stationary, a small object Q of mass 0.5 kg is placed on the turntable at a distance of 1.4 m from O . The turntable then begins to rotate, with a constant angular acceleration of \(1.12 \mathrm { rad } \mathrm { s } ^ { - 2 }\). Let \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) be the angular speed of the turntable. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-3_517_522_870_769} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Given that Q does not slip, find the components \(F _ { 1 }\) and \(F _ { 2 }\) of the frictional force acting on Q perpendicular and parallel to QO (see Fig. 2). Give your answers in terms of \(\omega\) where appropriate. The coefficient of friction between Q and the turntable is 0.65 .
   2. Find the value of \(\omega\) when Q is about to slip.
   3. Find the angle which the frictional force makes with QO when Q is about to slip.

3. \begin{figure}[h] \end{figure} A popular racket game involves a tennis ball of mass 0.1 kg which is attached to one end of a light inextensible string. The other end of the string is attached to the top of a fixed rigid pole. A boy strikes the ball such that it moves in a horizontal circle with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the string makes an angle of \(60 ^ { \circ }\) with the downward vertical as shown in Figure 1.

1. Find the tension in the string.
2. Find the length of the string.

5 On a fairground ride, the centre of a horizontal circular frame is attached to the top of a vertical pole, OP . When the frame and pole rotate, OP remains vertical and the frame remains horizontal. Chairs of mass 10 kg are attached to the frame by means of chains of length 2.5 m . The chains are modelled as being both light and inextensible. A side view of the situation when the ride is stationary is shown in Fig. 5. A chain fixed to point A on the circular frame supports a chair. The distance OA is 2 m . \begin{figure}[h] \end{figure} A child of mass 40 kg sits in a chair and, after a short time, the ride is rotating at a steady angular speed of \(\omega\) radians per second, with the chain inclined at an angle of \(50 ^ { \circ }\) to the downward vertical. The motion of the child and chair is in a horizontal circle.

1. Draw a sketch showing the forces acting on the chair when the ride is moving at this angular speed.
2. - Determine the tension in the chain.
   On another occasion, a man of mass 90 kg sits in the chair; after a short time, the ride is rotating in a horizontal circle at a steady speed of \(\omega\) radians per second, with the chain inclined at the same angle of \(50 ^ { \circ }\) to the downward vertical.
3. Without any detailed calculations, explain how your answers to part (b) for the child would compare with those for the man.
4. Explain why the chain is modelled as light.
5. State two other modelling assumptions that were used in answering part (b).

5. The diagram shows a fairground ride that consists of a number of seats suspended by chains that swing out as the centre rotates. \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-4_711_718_466_678} When the ride rotates at a constant angular speed of \(\omega = 1.4 \mathrm { rads } ^ { - 1 }\), the seats move in a horizontal circle with each chain making an angle \(\theta\) with the vertical. Each of the seats and the chains may be modelled as light. Assume that all chains have the same length and are inextensible. When a man of mass 75 kg occupies a seat, the tension in the chain is \(490 \sqrt { 3 } \mathrm {~N}\).

1. Show that \(\theta = 30 ^ { \circ }\).
2. Calculate the length of each chain.

1. The diagram below shows a particle \(P\), of mass 5 kg , attached to one end of a light inextensible string of length 3 m . The other end is fixed at a point \(A\). The particle \(P\) is moving in a horizonal circle with centre \(C\), where the point \(C\) is vertically below \(A\). The string is inclined at an angle \(\theta\) to the downward vertical, where \(\tan \theta = \frac { 20 } { 21 }\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-10_725_796_639_628}

Find the speed of the particle.

5 In this question use \(\boldsymbol { g } = 9.8 \mathbf { m ~ s } ^ { \mathbf { - 2 } }\).
A conical pendulum consists of a string of length 60 cm and a particle of mass 400 g . The string is at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-08_501_606_644_854} 5

1. Show that the tension in the string is 4.5 N . 5
2. Find the angular speed of the particle.
   [0pt] [3 marks]
   5
3. State two assumptions that you have made about the string.

2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point \(O\). The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of the circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_346_340_1580_842}

1. Show that the tension in the string is 22.6 N , correct to three significant figures.
2. Find the speed of the particle.

6 A particle is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle is set into motion, so that it describes a horizontal circle whose centre is vertically below \(O\). The angle between the string and the vertical is \(\theta\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-6_506_442_534_794}

1. The particle completes 40 revolutions every minute. Show that the angular speed of the particle is \(\frac { 4 \pi } { 3 }\) radians per second.
2. The radius of the circle is 0.2 metres. Find, in terms of \(\pi\), the magnitude of the acceleration of the particle.
3. The mass of the particle is \(m \mathrm {~kg}\) and the tension in the string is \(T\) newtons.
   1. Draw a diagram showing the forces acting on the particle.
   2. Explain why \(T \cos \theta = m g\).
   3. Find the value of \(\theta\), giving your answer to the nearest degree.

5 A particle, of mass 6 kg , is attached to one end of a light inextensible string. The other end of the string is attached to the fixed point \(O\). The particle is set in motion, so that it moves in a horizontal circle at constant speed, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of this circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-4_586_490_541_767} The particle moves in a horizontal circle with an angular speed of 40 revolutions per minute.

1. Show that the angular speed of the particle is \(\frac { 4 \pi } { 3 }\) radians per second.
2. Show that the tension in the string is 67.9 N , correct to three significant figures.
3. Find the radius of the horizontal circle.

6 A particle, of mass 4 kg , is attached to one end of a light inextensible string of length 1.2 metres. The other end of the string is attached to a fixed point \(O\). The particle moves in a horizontal circle at a constant speed. The angle between the string and the vertical is \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-4_529_554_1580_737}

1. Find the radius of the horizontal circle in terms of \(\theta\).
2. The angular speed of the particle is 5 radians per second. Find \(\theta\).

9 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-4_666_816_1384_662} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Show that however fast the particle travels \(A P\) will never become horizontal, and that the tension in the string is always greater than the weight of the particle.
2. Find the tension in the string in terms of \(m , l\) and \(\omega\).
3. Show that \(\omega ^ { 2 } h = g\) and calculate \(\omega\) when \(h\) is 0.5 m .

\includegraphics{figure_3} A particle of mass 0.24 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. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T\) N. The acceleration of the particle has magnitude \(7.5 \text{ m s}^{-2}\).

1. Show that \(\tan \theta = 0.75\) and find the value of \(T\). [4]
2. Find the speed of the particle. [2]