6.05c Horizontal circles: conical pendulum, banked tracks

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

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.

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.

1

1. A particle P of mass 1.5 kg is connected to a fixed point by a light inextensible string of length 3.2 m . The particle P is moving as a conical pendulum in a horizontal circle at a constant angular speed of \(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
   1. Find the tension in the string.
   2. Find the angle that the string makes with the vertical.
2. A particle Q of mass \(m\) moves on a smooth horizontal surface, and is connected to a fixed point on the surface by a light elastic string of natural length \(d\) and stiffness \(k\). With the string at its natural length, Q is set in motion with initial speed \(u\) perpendicular to the string. In the subsequent motion, the maximum length of the string is \(2 d\), and the string first returns to its natural length after time \(t\). You are given that \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) and \(t = A k ^ { \alpha } d ^ { \beta } m ^ { \gamma }\), where \(A\) is a dimensionless constant.
   1. Show that the dimensions of \(k\) are \(\mathrm { MT } ^ { - 2 }\).
   2. Show that the equation \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) is dimensionally consistent.
   3. Find \(\alpha , \beta\) and \(\gamma\). You are now given that Q has mass 5 kg , and the string has natural length 0.7 m and stiffness \(60 \mathrm { Nm } ^ { - 1 }\).
   4. Find the initial speed \(u\), and use conservation of energy to find the speed of Q at the instant when the length of the string is double its natural length.

2

1. The fixed point A is vertically above the fixed point B . A light inextensible string of length 5.4 m has one end attached to A and the other end attached to B. The string passes through a small smooth ring R of mass 0.24 kg , and R is moving at constant angular speed in a horizontal circle. The circle has radius 1.6 m , and \(\mathrm { AR } = 3.4 \mathrm {~m} , \mathrm { RB } = 2.0 \mathrm {~m}\), as shown in Fig. 2 . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-3_565_504_447_753} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Find the tension in the string.
   2. Find the angular speed of R .
2. A particle P of mass 0.3 kg is joined to a fixed point O by a light inextensible string of length 1.8 m . The particle P moves without resistance in part of a vertical circle with centre O and radius 1.8 m . When OP makes an angle of \(25 ^ { \circ }\) with the downward vertical, the tension in the string is 15 N .
   1. Find the speed of P when OP makes an angle of \(25 ^ { \circ }\) with the downward vertical.
   2. Find the tension in the string when OP makes an angle of \(60 ^ { \circ }\) with the upward vertical.
   3. Find the speed of P at the instant when the string becomes slack.

6. \begin{figure}[h] \end{figure} The two ends of a light inextensible string of length \(3 a\) are attached to fixed points \(Q\) and \(R\) which are a distance of \(a \sqrt { } 3\) apart with \(R\) vertically below \(Q\). A particle \(P\) of mass \(m\) is attached to the string at a distance of \(2 a\) from \(Q\). \(P\) is given a horizontal speed, \(u\), such that it moves in a horizontal circle with both sections of the string taut as shown in Figure 3.

1. Show that \(\angle P R Q\) is a right angle.
2. Find \(\angle P Q R\) in degrees.
3. Find, in terms of \(a , g , m\) and \(u\), the tension in the section of string
   1. \(P Q\),
   2. \(P R\).
4. Show that \(u ^ { 2 } \geq \frac { g a } { \sqrt { 3 } }\).

6. A car is travelling on a horizontal racetrack round a circular bend of radius 40 m . The coefficient of friction between the car and the road is \(\frac { 2 } { 5 }\).

1. Find the maximum speed at which the car can travel round the bend without slipping, giving your answer correct to 3 significant figures.
   (5 marks)
   The owner of the track decides to bank the corner at an angle of \(25 ^ { \circ }\) in order to enable the cars to travel more quickly.
2. Show that this increases the maximum speed at which the car can travel round the bend without slipping by 63\%, correct to the nearest whole number.
   (8 marks)

3. A coin of mass 5 grams is placed on a vinyl disc rotating on a record player. The distance between the centre of the coin and the centre of the disc is 0.1 m and the coefficient of friction between the coin and the disc is \(\mu\). The disc rotates at 45 revolutions per minute around a vertical axis at its centre and the coin moves with it and does not slide. By modelling the coin as a particle and giving your answers correct to an appropriate degree of accuracy, find

1. the speed of the coin,
2. the horizontal and vertical components of the force exerted on the coin by the disc. Given that the coin is on the point of moving,
3. show that, correct to 2 significant figures, \(\mu = 0.23\).

7. A cyclist is travelling round a circular bend of radius 25 m on a track which is banked at an angle of \(35 ^ { \circ }\) to the horizontal. In a model of the situation, the cyclist and her bicycle are represented by a particle of mass 60 kg and air resistance and friction are ignored. Using this model and assuming that the cyclist is not slipping,

1. find, correct to 3 significant figures, the speed at which she is travelling. In tests it is found that the cyclist must travel at a minimum speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to prevent the bicycle from slipping down the slope. A more refined model is now used with a coefficient of friction between the bicycle and the track of \(\mu\). Using this model,
2. show that \(\mu = 0.227\), correct to 3 significant figures,
3. find, correct to 2 significant figures, the maximum speed at which the cyclist can travel without slipping up the slope. END

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.

1. A car moves round a bend which is banked at a constant angle of \(\theta ^ { \circ }\) to the horizontal.

When the car is travelling at a constant speed of \(80 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius 500 m .

1. Find the value of \(\theta\).
2. Identify one limitation of this model. The speed of the car is increased so that it is now travelling at a constant speed of \(90 \mathrm { kmh } ^ { - 1 }\) The car is still modelled as a particle moving in a horizontal circle of radius 500 m .
3. Describe the extra force that will now be acting on the car, stating the direction of this force.

   VILU SIHI NI IIIUM ION OC

   VGHV SIHILNI IMAM ION OO

   VJYV SIHI NI JIIYM ION OC

1. A light inextensible string has length \(8 a\). One end of the string is attached to a fixed point \(A\) and the other end of the string is attached to a fixed point \(B\), with \(A\) vertically above \(B\) and \(A B = 4 a\). A small ball of mass \(m\) is attached to a point \(P\) on the string, where \(A P = 5 a\).

The ball moves in a horizontal circle with constant speed \(v\), with both \(A P\) and \(B P\) taut.
The string will break if the tension in it exceeds \(\frac { 3 m g } { 2 }\) By modelling the ball as a particle and assuming the string does not break,

1. show that \(\frac { 9 a g } { 4 } < v ^ { 2 } \leqslant \frac { 27 a g } { 4 }\)
2. find the least possible time needed for the ball to make one complete revolution.