6.05b Circular motion: v=r*omega and a=v^2/r

5 Two particles, \(A\) and \(B\), are connected by a light inextensible string which passes through a hole in a smooth horizontal table. The edges of the hole are also smooth. Particle \(A\), of mass 1.4 kg , moves, on the table, with constant speed in a circle of radius 0.3 m around the hole. Particle \(B\), of mass 2.1 kg , hangs in equilibrium under the table, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-4_684_1022_1176_504}

1. Find the angular speed of particle \(A\).
2. Find the speed of particle \(A\).
3. Find the time taken for particle \(A\) to complete one full circle around the hole.

5 Tom is travelling on a train which is moving at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal track. Tom has placed his mobile phone on a rough horizontal table. The coefficient of friction between the phone and the table is 0.2 . The train moves round a bend of constant radius. The phone does not slide as the train travels round the bend. Model the phone as a particle moving round part of a circle, with centre \(O\) and radius \(r\) metres. Find the least possible value of \(r\).

4 A particle, \(P\), of mass 5 kg is attached to two light inextensible strings, \(A P\) and \(B P\). The other ends of the strings are attached to the fixed points \(A\) and \(B\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in a horizontal circle of radius 0.6 metres with centre \(B\). The string \(A P\) is inclined at \(20 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut when the particle is moving. \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-08_835_568_568_719}

1. Find the tension in the string \(A P\).
2. The speed of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the tension, \(T _ { B P }\), in the string \(B P\) is given by $$T _ { B P } = \frac { 25 } { 3 } v ^ { 2 } - 5 g \tan 20 ^ { \circ }$$
3. Find \(v\) when the tensions in the two strings are equal.

5 An item of clothing is placed inside a washing machine. The drum of the washing machine has radius 30 cm and rotates, about a fixed horizontal axis, at a constant angular speed of 900 revolutions per minute. Model the item of clothing as a particle of mass 0.8 kg and assume that the clothing travels in a vertical circle with constant angular speed. Find the minimum magnitude of the normal reaction force exerted by the drum on the clothing and find the maximum magnitude of the normal reaction force exerted by the drum on the clothing.
[0pt] [6 marks]

4 A light inextensible string of length 0.6 m has one end fixed to a point \(A\) on a smooth horizontal plane. The other end of the string is attached to a particle \(B\), of mass 0.4 kg , which rotates about \(A\) with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) on the surface of the plane.

1. Calculate the tension in the string. A particle \(P\) of mass 0.1 kg is attached to the mid-point of the string. The line \(A P B\) is straight and rotation continues at \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Calculate the tension in the section of the string \(A P\).
3. Calculate the total kinetic energy of the system.

4 A smooth cylinder of radius \(a \mathrm {~m}\) is fixed with its axis horizontal and \(O\) is the centre of a cross-section. Particle \(P\), of mass 0.4 kg , and particle \(Q\), of mass 0.6 kg , are connected by a light inextensible string of length \(\pi a \mathrm {~m}\). The string is held at rest with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the crosssection 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). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 3.92 a ( 3 \theta - 2 \sin \theta )\) and find an expression in terms of \(\theta\) for the normal force of the cylinder on \(P\) at this time.
2. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(\sin \alpha = k \alpha\) where \(k\) is a constant to be found.

7 \includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-4_517_677_267_733} A particle \(P\) of mass \(m \mathrm {~kg}\) is slightly disturbed from rest at the highest point on the surface of a smooth fixed sphere of radius \(a\) m and centre \(O\). The particle starts to move downwards on the surface. While \(P\) remains on the surface \(O P\) makes an angle of \(\theta\) radians with the upward vertical and has angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram). The sphere exerts a force of magnitude \(R \mathrm {~N}\) on \(P\).

1. Show that \(a \omega ^ { 2 } = 2 g ( 1 - \cos \theta )\).
2. Find an expression for \(R\) in terms of \(m , g\) and \(\theta\). At the instant that \(P\) loses contact with the surface of the sphere, find
3. the transverse component of the acceleration of \(P\),
4. the rate of change of \(R\) with respect to time \(t\), in terms of \(m , g\) and \(a\).

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 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. 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 \includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-4_783_783_255_641} A uniform circular disc with centre \(C\) has mass \(m\) and radius \(a\). The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point \(A\) on the disc, where \(A C = \frac { 1 } { 2 } a\). The disc is slightly disturbed from rest in the position with \(C\) vertically above \(A\). When \(A C\) makes an angle \(\theta\) with the upward vertical the force exerted by the axis on the disc has components \(R\) parallel to \(A C\) and \(S\) perpendicular to \(A C\) (see diagram).

1. Show that the angular speed of the disc is \(\sqrt { \frac { 4 g ( 1 - \cos \theta ) } { 3 a } }\).
2. Find the angular acceleration of the disc, in terms of \(a , g\) and \(\theta\).
3. Find \(R\) and \(S\), in terms of \(m , g\) and \(\theta\).
4. Find the magnitude of the force exerted by the axis on the disc at an instant when \(R = 0\).

1 A camshaft inside an engine is rotating with angular speed \(42 \mathrm { rads } ^ { - 1 }\). When the throttle is opened the camshaft speeds up with constant angular acceleration, and 8 seconds after the throttle was opened the angular speed is \(76 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the angular acceleration of the camshaft.
2. Find the time taken for the camshaft to turn through 810 radians from the moment that the throttle was opened.

6 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-4_640_608_267_715} A smooth wire forms a circle with centre \(O\) and radius \(a\), and is fixed in a vertical plane. The highest point on the wire is \(A\). A small ring \(R\) of mass \(m\) moves along the wire. A light elastic string, with natural length \(\frac { 1 } { 2 } a\) and modulus of elasticity \(2 m g\), has one end attached to \(A\) and the other end attached to \(R\). The string \(A R\) makes an angle \(\theta\) (measured anticlockwise) with the downward vertical (see diagram), and you may assume that the string does not become slack.

1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy of the system is \(m g a \left( 6 \cos ^ { 2 } \theta - 4 \cos \theta + \frac { 1 } { 2 } \right)\).
2. Show that there are two positions of equilibrium for which \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).
3. For each of these positions of equilibrium, determine whether it is stable or unstable.

7 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-5_584_686_264_678} \(A B C D\) is a uniform rectangular lamina with mass \(m\) and sides \(A B = 6 a\) and \(A D = 8 a\). The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through \(A\), and it is released from rest in the position with \(D\) vertically above \(A\). When the diagonal \(A C\) makes an angle \(\theta\) below the horizontal, the force acting on the lamina at \(A\) has components \(R\) parallel to \(C A\) and \(S\) perpendicular to \(C A\) (see diagram).

1. Find the moment of inertia of the lamina about the axis through \(A\), in terms of \(m\) and \(a\).
2. Show that the angular speed of the lamina is \(\sqrt { \frac { 3 g ( 4 + 5 \sin \theta ) } { 50 a } }\).
3. Find the angular acceleration of the lamina, in terms of \(a , g\) and \(\theta\).
4. Find \(R\) and \(S\), in terms of \(m , g\) and \(\theta\).

2 A uniform solid circular cone has mass \(M\) and base radius \(R\).

1. Show by integration that the moment of inertia of the cone about its axis of symmetry is \(\frac { 3 } { 10 } M R ^ { 2 }\). (You may assume the standard formula \(\frac { 1 } { 2 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about its axis and that the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).) The axis of symmetry of the cone is fixed vertically and the cone is rotating about its axis at an angular speed of \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A frictional couple of constant moment 0.027 Nm is applied to the cone bringing it to rest. Given that the mass of the cone is 2 kg and its base radius is 0.3 m , find
2. the constant angular deceleration of the cone,
3. the time taken for the cone to come to rest from the instant that the couple is applied.

4 A uniform square lamina has mass \(m\) and sides of length \(2 a\).

1. Calculate the moment of inertia of the lamina about an axis through one of its corners perpendicular to its plane. \includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-3_693_640_434_715} The uniform square lamina has centre \(C\) and is free to rotate in a vertical plane about a fixed horizontal axis passing through one of its corners \(A\). The lamina is initially held such that \(A C\) is vertical with \(C\) above \(A\). The lamina is slightly disturbed from rest from this initial position. When \(A C\) makes an angle \(\theta\) with the upward vertical, the force exerted by the axis on the lamina has components \(X\) parallel to \(A C\) and \(Y\) perpendicular to \(A C\) (see diagram).
2. Show that the angular speed, \(\omega\), of the lamina satisfies \(a \omega ^ { 2 } = \frac { 3 } { 4 } g \sqrt { 2 } ( 1 - \cos \theta )\).
3. Find \(X\) and \(Y\) in terms of \(m , g\) and \(\theta\). \section*{Question 5 begins on page 4.}
   \includegraphics[max width=\textwidth, alt={}]{639c658e-0aca-4161-9e77-0f4c494b0b55-4_767_337_248_863}
   A pendulum consists of a uniform rod \(A B\) of length \(4 a\) and mass \(4 m\) and a spherical shell of radius \(a\), mass \(m\) and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the surface of the shell in such a way that \(A B C\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\) (see diagram).
4. Show that the moment of inertia of the pendulum about the axis of rotation is \(47 m a ^ { 2 }\). A particle of mass \(m\) is moving horizontally in the plane in which the pendulum is free to rotate. The particle has speed \(\sqrt { k g a }\), where \(k\) is a positive constant, and strikes the rod at a distance \(3 a\) from \(A\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(P\) starts to rotate.
5. Show that the initial angular speed of \(P\) is \(\frac { 3 } { 56 } \sqrt { \frac { k g } { a } }\).
6. For the case \(k = 4\), find the angle that \(P\) has turned through when \(P\) first comes to instantaneous rest.
7. Find the least value of \(k\) such that the rod reaches the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-5_437_903_269_573} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\sqrt { 3 } m g\) is attached to \(A\). The string passes over a small smooth fixed pulley \(C\), where \(A C\) is horizontal and \(A C = a\). The other end of the string is attached to the rod at its mid-point \(D\). The rod makes an angle \(\theta\) below the horizontal (see diagram).
8. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = m g a ( \sqrt { 3 } - \sin \theta - \sqrt { 3 } \cos \theta ) .$$
9. Show that \(\theta = \frac { 1 } { 6 } \pi\) is a position of stable equilibrium for the system. The system is making small oscillations about the equilibrium position.
10. By differentiating the energy equation with respect to time, show that $$\frac { 4 } { 3 } a \ddot { \theta } = g ( \cos \theta - \sqrt { 3 } \sin \theta ) .$$
11. Using the substitution \(\theta = \phi + \frac { 1 } { 6 } \pi\), show that the motion is approximately simple harmonic, and find the approximate period of the oscillations. \section*{END OF QUESTION PAPER}

1 A turntable is rotating at \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). The turntable is then accelerated so that after 4 revolutions it is rotating at \(12.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Assuming that the angular acceleration of the turntable is constant,

1. find the angular acceleration,
2. find the time taken to increase its angular speed from \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(12.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

6 A pendulum consists of a uniform rod \(A B\) of length \(2 a\) and mass \(2 m\) and a particle of mass \(m\) that is attached to the end \(B\). The pendulum can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\).

1. Show that the moment of inertia of this pendulum about the axis of rotation is \(\frac { 20 } { 3 } m a ^ { 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_572_86_852_575} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_582_456_842_1050} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The pendulum is initially held with \(B\) vertically above \(A\) (see Fig.1) and it is slightly disturbed from this position. When the angle between the pendulum and the upward vertical is \(\theta\) radians the pendulum has angular speed \(\omega \mathrm { rads } ^ { - 1 }\) (see Fig. 2).
2. Show that $$\omega ^ { 2 } = \frac { 6 g } { 5 a } ( 1 - \cos \theta ) .$$
3. Find the angular acceleration of the pendulum in terms of \(g , a\) and \(\theta\). At an instant when \(\theta = \frac { 1 } { 3 } \pi\), the force acting on the pendulum at \(A\) has magnitude \(F\).
4. Find \(F\) in terms of \(m\) and \(g\). It is given that \(a = 0.735 \mathrm {~m}\).
5. Show that the time taken for the pendulum to move from the position \(\theta = \frac { 1 } { 6 } \pi\) to the position \(\theta = \frac { 1 } { 3 } \pi\) is given by $$k \int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta ,$$ stating the value of the constant \(k\). Hence find the time taken for the pendulum to rotate between these two points. (You may quote an appropriate result given in the List of Formulae (MF1).) \section*{END OF QUESTION PAPER}

1. A uniform circular disc, of mass \(m\) and radius \(a\), is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) is a tangent to the disc at the point \(A\). The centre \(O\) of the disc moves in a vertical plane that is perpendicular to \(L\).

The disc is held at rest with its plane horizontal and released.

1. Find the angular acceleration of the disc when it has turned through an angle of \(\frac { \pi } { 3 }\)
2. Find the magnitude of the component, in a direction perpendicular to the disc, of the force of the axis \(L\) acting on the disc at \(A\), when the disc has turned through an angle of \(\frac { \pi } { 3 }\)

7. (a) Find, using integration, the moment of inertia of a uniform solid hemisphere, of mass \(m\) and radius \(a\), about a diameter of its plane face.
[0pt] [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about a diameter is \(\frac { 1 } { 4 } m r ^ { 2 }\).]
(b) Hence find the moment of inertia of a uniform solid sphere, of mass \(M\) and radius \(a\), about a diameter.

4. Find, using integration, the moment of inertia of a uniform cylindrical shell of radius \(r\), height \(h\) and mass \(M\), about a diameter of one end.
(10)

5. \begin{figure}[h] \end{figure} A uniform piece of wire \(A B C\), of mass \(2 m\) and length \(4 a\), is bent into two straight equal portions, \(A B\) and \(B C\), which are at right angles to each other, as shown in Figure 1. The wire rotates freely in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the wire.

1. Show that the moment of inertia of the wire about \(L\) is \(\frac { 20 m a ^ { 2 } } { 3 }\)
2. By writing down an equation of rotational motion for the wire as it rotates about \(L\), find the period of small oscillations of the wire about its position of stable equilibrium.

7. A uniform square lamina \(P Q R S\), of mass \(m\) and side \(2 a\), is free to rotate about a fixed smooth horizontal axis which passes through \(P\) and \(Q\). The lamina hangs at rest in a vertical plane with \(S R\) below \(P Q\) and is given a horizontal impulse of magnitude \(J\) at the midpoint of \(S R\). The impulse is perpendicular to \(S R\).

1. Find the initial angular speed of the lamina.
2. Find the magnitude of the angular deceleration of the lamina at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians.
3. Find the magnitude of the component of the force exerted on the lamina by the axis, in a direction perpendicular to the lamina, at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians. \includegraphics[max width=\textwidth, alt={}, center]{f932d7cb-1299-41d1-8248-cfbf639795ed-12_2255_50_315_1978}