6.05d Variable speed circles: energy methods

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\). Show that the least possible value of \(u\) is \(\sqrt{(ag)}\). [2] It is now given that \(u = \sqrt{(ag)}\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac{1}{4}m\). Find the speed of the combined particle immediately after the collision. [4] In the subsequent motion, when \(OP\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m\), \(g\) and \(\theta\). [5] Find the value of \(\cos \theta\) when the string becomes slack. [2]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\) and the point \(C\) is on the inner surface of the sphere, vertically below \(O\). The points \(A\) and \(B\) on the inner surface of the sphere are the ends of a diameter of the sphere. The diameter \(AOB\) makes an acute angle \(\alpha\) with the vertical, where \(\cos \alpha = \frac{4}{5}\), with \(A\) below the horizontal level of \(B\). The particle is projected from \(A\) with speed \(u\), and moves along the inner surface of the sphere towards \(C\). The normal reaction forces on the particle at \(A\) and \(C\) are in the ratio \(8 : 9\).

1. Show that \(u^2 = 4ag\). [6]
2. Determine whether \(P\) reaches \(B\) without losing contact with the inner surface of the sphere. [6]

OR A machine is used to produce metal rods. When the machine is working efficiently, the lengths, \(x\) cm, of the rods have a normal distribution with mean 150 cm and standard deviation 1.2 cm. The machine is checked regularly by taking random samples of 200 rods. The latest results are shown in the following table.

Interval	\(146 \leqslant x < 147\)	\(147 \leqslant x < 148\)	\(148 \leqslant x < 149\)	\(149 \leqslant x < 150\)
Observed frequency	1	2	23	52

\(150 \leqslant x < 151\)	\(151 \leqslant x < 152\)	\(152 \leqslant x < 153\)	\(153 \leqslant x < 154\)
69	36	15	2

As a first check, the sample is used to calculate an estimate for the mean.

1. Show that an estimate for the mean from this sample is close to 150 cm. [2]

As a second check, the results are tested for goodness of fit of the normal distribution with mean 150 cm and standard deviation 1.2 cm. The relevant expected frequencies, found using the normal distribution function given in the List of Formulae (MF10), are shown in the following table.

Interval	\(x < 147\)	\(147 \leqslant x < 148\)	\(148 \leqslant x < 149\)	\(149 \leqslant x < 150\)
Observed frequency	1	2	23	52
Expected frequency	1.24	8.32	30.94	59.50

\(150 \leqslant x < 151\)	\(151 \leqslant x < 152\)	\(152 \leqslant x < 153\)	\(153 \leqslant x\)
69	36	15	2
59.50	30.94	8.32	1.24

1. Show how the expected frequency for \(151 \leqslant x < 152\) is obtained. [3]
2. Test, at the 5\% significance level, the goodness of fit of the normal distribution to the results. [7]

\includegraphics{figure_3} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The point \(A\) is such that \(OA = a\) and \(OA\) makes an angle \(\alpha\) with the upward vertical, where \(\tan \alpha = \frac{15}{8}\). The particle is projected downwards from \(A\) with speed \(u\) perpendicular to the string and moves in a vertical plane (see diagram). The string becomes slack after the string has rotated through \(270°\) from its initial position, with the particle now at the point \(B\). \begin{enumerate}[label=(\roman*)] \item Show that \(u^2 = 2ag\). [5] \item Find the maximum tension in the string as the particle moves from \(A\) to \(B\). [4] \end{enumerate]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). When the particle is hanging vertically below \(O\), it is projected horizontally with speed \(u\) so that it begins to move along a circular path. When \(P\) is at the lowest point of its motion, the tension in the string is \(T\). When \(OP\) makes an angle \(\theta\) with the upward vertical, the tension in the string is \(S\).

1. Show that \(S = T - 3mg(1 + \cos\theta)\). [5]
2. Given that \(u = \sqrt{4ag}\), find the value of \(\cos\theta\) when the string goes slack. [2]

\includegraphics{figure_2} A particle \(P\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held with the string taut and making an angle \(\theta\) with the downward vertical. The particle \(P\) is then projected with speed \(\frac{1}{2}\sqrt{5ag}\) perpendicular to the string and just completes a vertical circle (see diagram). Find the value of \(\cos\theta\). [5]

A particle \(P\), of mass \(m\), is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) moves in complete vertical circles about \(O\) with the string taut. The points \(A\) and \(B\) are on the path of \(P\) with \(AB\) a diameter of the circle. \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(\sqrt{5ag}\). The ratio of the tension in the string when \(P\) is at \(A\) to the tension in the string when \(P\) is at \(B\) is \(9 : 5\).

1. Find the value of \(\cos \theta\). [6]
2. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) during its motion. [2]

One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The particle \(P\) is held vertically below \(O\) with the string taut and then projected horizontally. When the string makes an angle of \(60°\) with the upward vertical, \(P\) becomes detached from the string. In its subsequent motion, \(P\) passes through the point \(A\) which is a distance \(a\) vertically above \(O\).

1. The speed of \(P\) when it becomes detached from the string is \(V\). Use the equation of the trajectory of a projectile to find \(V\) in terms of \(a\) and \(g\). [4]
2. Find, in terms of \(m\) and \(g\), the tension in the string immediately after \(P\) is initially projected horizontally. [4]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The string is held taut with \(OP\) making an angle \(\alpha\) with the downward vertical, where \(\cos \alpha = \frac{2}{3}\). The particle \(P\) is projected perpendicular to \(OP\) in an upwards direction with speed \(\sqrt{3ag}\). It then starts to move along a circular path in a vertical plane. Find the cosine of the angle between the string and the upward vertical when the string first becomes slack. [4]

\includegraphics{figure_5} A bead of mass \(m\) moves on a smooth circular wire, with centre \(O\) and radius \(a\), in a vertical plane. The bead has speed \(v_A\) when it is at the point \(A\) where \(OA\) makes an angle \(\alpha\) with the downward vertical through \(O\), and \(\cos \alpha = \frac{2}{3}\). Subsequently the bead has speed \(v_B\) at the point \(B\), where \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). Angle \(AOB\) is a right angle (see diagram). The reaction of the wire on the bead at \(B\) is in the direction \(OB\) and has magnitude equal to \(\frac{1}{6}\) of the magnitude of the reaction when the bead is at \(A\).

1. Find, in terms of \(m\) and \(g\), the magnitude of the reaction at \(B\). [6]
2. Given that \(v_A = \sqrt{kag}\), find the value of \(k\). [2]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\) with the string taut. It is given that \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\tan\theta = \frac{3}{4}\). The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{5ag}\), and it starts to move along a circular path in a vertical plane. When \(P\) is at the point \(B\), where angle \(AOB\) is a right angle, the tension in the string is \(T\). Find \(T\) in terms of \(m\) and \(g\). [5]

\includegraphics{figure_6} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held with the string taut and the string makes an angle \(\theta\) with the downward vertical through \(O\). The particle is projected at right angles to the string with speed \(\frac{1}{3}\sqrt{10ag}\) and begins to move downwards along a circular path. When the string is vertical, it strikes a small smooth peg at the point \(A\) which is vertically below \(O\). The circular path and the point \(A\) are in the same vertical plane. After the string strikes the peg, the particle \(P\) begins to move in a vertical circle with centre \(A\). When the string makes an angle \(\theta\) with the upward vertical through \(A\) the string becomes slack (see diagram). The distance of \(A\) below \(O\) is \(\frac{5}{6}a\).

1. Find the value of \(\cos \theta\). [6]
2. Find the ratio of the tensions in the string immediately before and immediately after it strikes the peg. [4]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\) with the string taut. It is given that \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\tan \theta = \frac{3}{4}\). The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{5ag}\), and it starts to move along a circular path in a vertical plane. When \(P\) is at the point \(B\), where angle \(AOB\) is a right angle, the tension in the string is \(T\). Find \(T\) in terms of \(m\) and \(g\). [5]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\).

1. \includegraphics{figure_5a} The particle \(P\) moves in a horizontal circle with a constant angular speed \(\omega\) with the string inclined at \(60°\) to the downward vertical through \(O\) (see diagram). Show that \(\omega^2 = \frac{2g}{a}\). [4]
2. The particle now hangs at rest and is then projected horizontally so that it begins to move in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the downward vertical through \(O\), the angular speed of \(P\) is \(\sqrt{\frac{2g}{a}}\). The string first goes slack when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\). [6]

\includegraphics{figure_3} Figure 3 shows a fixed hollow sphere of internal radius \(a\) and centre \(O\). A particle \(P\) of mass \(m\) is projected horizontally from the lowest point \(A\) of a sphere with speed \(\sqrt{\left(\frac{5}{4}ag\right)}\). It moves in a vertical circle, centre \(O\), on the smooth inner surface of the sphere. The particle passes through the point \(B\), which is in the same horizontal plane as \(O\). It leaves the surface of the sphere at the point \(C\), where \(OC\) makes an angle \(\theta\) with the upward vertical.

1. Find, in terms of \(m\) and \(g\), the normal reaction between \(P\) and the surface of the sphere at \(B\). [4]
2. Show that \(\theta = 60°\). [7]

After leaving the surface of the sphere, \(P\) meets it again at the point \(A\).

1. Find, in terms of \(a\) and \(g\), the time \(P\) takes to travel from \(C\) to \(A\). [4]

\includegraphics{figure_6} A trapeze artiste of mass 60 kg is attached to the end \(A\) of a light inextensible rope \(OA\) of length 5 m. The artiste must swing in an arc of a vertical circle, centre \(O\), from a platform \(P\) to another platform \(Q\), where \(PQ\) is horizontal. The other end of the rope is attached to the fixed point \(O\) which lies in the vertical plane containing \(PQ\), with \(\angle POQ = 120^{\circ}\) and \(OP = OQ = 5\) m, as shown in Figure 6. As part of her act, the artiste projects herself from \(P\) with speed \(\sqrt{15}\) m s\(^{-1}\) in a direction perpendicular to the rope \(OA\) and in the plane \(POQ\). She moves in a circular arc towards \(Q\). At the lowest point of her path she catches a ball of mass \(m\) kg which is travelling towards her with speed 3 m s\(^{-1}\) and parallel to \(QP\). After catching the ball, she comes to rest at the point \(Q\). By modelling the artiste and the ball as particles and ignoring her air resistance, find

1. the speed of the artiste immediately before she catches the ball, [4]
2. the value of \(m\), [7]
3. the tension in the rope immediately after she catches the ball. [3]

\includegraphics{figure_1} A smooth solid hemisphere, of radius 0.8 m and centre \(O\), is fixed with its plane face on a horizontal table. A particle of mass 0.5 kg is projected horizontally with speed \(u\) m s\(^{-1}\) from the highest point \(A\) of the hemisphere. The particle leaves the hemisphere at the point \(B\), which is a vertical distance of 0.2 m below the level of \(A\). The speed of the particle at \(B\) is \(v\) m s\(^{-1}\) and the angle between \(OA\) and \(OB\) is \(\theta\), as shown in Fig. 1.

1. Find the value of \(\cos \theta\). [1]
2. Show that \(v^2 = 5.88\). [3]
3. Find the value of \(u\). [3]

A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A particle, of mass \(m\), is attached to the other end of the string. The particle is moving in a vertical circle with centre \(O\). The point \(Q\) is the highest point of the particle's path. When the particle is at \(P\), vertically below \(O\), the string is taut and the particle is moving with speed \(7\sqrt{ag}\), as shown in the diagram. \includegraphics{figure_5}

1. Find, in terms of \(g\) and \(a\), the speed of the particle at the point \(Q\). [4 marks]
2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(Q\). [3 marks]

A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(R\) vertically below \(O\). The particle is set into motion with a horizontal velocity \(u\) so that it moves in a complete vertical circle with centre \(O\). The point \(T\) on the circle is such that angle \(ROT\) is \(30°\), as shown in the diagram. \includegraphics{figure_5}

1. Find, in terms of \(g\), \(l\) and \(u\), the speed of the particle at the point \(T\). [3 marks]
2. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle is at the point \(T\). [3 marks]
3. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle returns to the point \(R\). [2 marks]
4. The particle makes complete revolutions. Find, in terms of \(g\) and \(l\), the minimum value of \(u\). [4 marks]

A particle \(P\) is attached to one end of a light inextensible string of length \(l\) m. The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is given a horizontal speed \(u\) ms\(^{-1}\) and starts to move in a vertical circle. Given that the string becomes slack when it makes an angle of 120° with the downward vertical through \(O\),

1. show that \(u^2 = \frac{7gl}{2}\). [10 marks]
2. Find, in terms of \(l\), the greatest height above \(O\) reached by \(P\) in the subsequent motion. [7 marks]

\includegraphics{figure_7} A hollow cylinder has internal radius \(a\). The cylinder is fixed with its axis horizontal. A particle \(P\) of mass \(m\) is at rest in contact with the smooth inner surface of the cylinder. \(P\) is given a horizontal velocity \(u\), in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While \(P\) remains in contact with the surface, \(OP\) makes an angle \(\theta\) with the downward vertical, where \(O\) is the centre of the circle. The speed of \(P\) is \(v\) and the magnitude of the force exerted on \(P\) by the surface is \(R\) (see diagram).

1. Find \(v^2\) in terms of \(u\), \(a\), \(g\) and \(\theta\) and show that \(R = \frac{mu^2}{a} + mg(3\cos\theta - 2)\). [7]
2. Given that \(P\) just reaches the highest point of the circle, find \(u^2\) in terms of \(a\) and \(g\), and show that in this case the least value of \(v^2\) is \(ag\). [4]
3. Given instead that \(P\) oscillates between \(\theta = \pm\frac{1}{5}\pi\) radians, find \(u^2\) in terms of \(a\) and \(g\). [2]

\includegraphics{figure_7} A particle \(P\) is attached to a fixed point \(O\) by a light inextensible string of length \(0.7\) m. A particle \(Q\) is in equilibrium suspended from \(O\) by an identical string. With the string \(OP\) taut and horizontal, \(P\) is projected vertically downwards with speed \(6\) m s\(^{-1}\) so that it strikes \(Q\) directly (see diagram). \(P\) is brought to rest by the collision and \(Q\) starts to move with speed \(4.9\) m s\(^{-1}\).

1. Find the speed of \(P\) immediately before the collision. Hence find the coefficient of restitution between \(P\) and \(Q\). [3]
2. Given that the speed of \(Q\) is \(v\) m s\(^{-1}\) when \(OQ\) makes an angle \(\theta\) with the downward vertical, find an expression for \(v^2\) in terms of \(\theta\), and show that the tension in the string \(OQ\) is \(14.7m(1 + 2\cos\theta)\) N, where \(m\) kg is the mass of \(Q\). [6]
3. Find the radial and transverse components of the acceleration of \(Q\) at the instant that the string \(OQ\) becomes slack. [4]
4. Show that \(V^2 = 0.8575\), where \(V\) m s\(^{-1}\) is the speed of \(Q\) when it reaches its greatest height (after the string \(OQ\) becomes slack). Hence find the greatest height reached by \(Q\) above its initial position. [4]

One end of a light inextensible string of length \(0.8\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.5\) kg is attached to the other end of the string. \(P\) is projected horizontally from the point \(0.8\) m vertically below \(O\) with speed \(5.6\) m s\(^{-1}\). \(P\) starts to move in a vertical circle with centre \(O\). The speed of \(P\) is \(v\) m s\(^{-1}\) when the string makes an angle \(\theta\) with the downward vertical.

1. While the string remains taut, show that \(v^2 = 15.68(1 + \cos \theta)\), and find the tension in the string in terms of \(\theta\). [7]
2. For the instant when the string becomes slack, find the value of \(\theta\) and the value of \(v\). [3]
3. Find, in either order, the speed of \(P\) when it is at its greatest height after the string becomes slack, and the greatest height reached by \(P\) above its point of projection. [4]

\includegraphics{figure_7} 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. \(P\) is projected horizontally from the point \(0.5\) m below \(O\) with speed \(u\text{ ms}^{-1}\). When the string makes an angle of \(\theta\) with the downward vertical the particle has speed \(v\text{ ms}^{-1}\) (see diagram).

1. Show that, while the string is taut, the tension, \(T\) N, in the string is given by $$T = 5.88\cos \theta + 0.4u^2 - 3.92.$$ [5]
2. Find the least value of \(u\) for which the particle will move in a complete circle. [3]
3. If in fact \(u = 3.5\text{ ms}^{-1}\), find the speed of the particle at the point where the string first becomes slack. [4]

END OF QUESTION PAPER

\includegraphics{figure_5} One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs at rest. \(P\) is then projected horizontally from this position with speed \(2\sqrt{ag}\). When the string makes an angle \(\theta\) with the upward vertical \(P\) has speed \(v\) (see diagram). The tension in the string is \(T\).

1. Find an expression for \(T\) in terms of \(m\), \(g\) and \(\theta\), and hence find the height of \(P\) above its initial level when the string becomes slack. [6]

\(P\) is now projected horizontally from the same initial position with speed \(U\).

1. Find the set of values of \(U\) for which the string does not remain taut in the subsequent motion. [5]

A uniform rectangular lamina \(ABCD\) has mass 20 kg and sides of lengths \(AB = 0.6\) m and \(BC = 1.8\) m. It rotates in its own vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the mid-point of \(AB\).

1. Show that the moment of inertia of the lamina about the axis is 22.2 kg m\(^2\). [3]

\includegraphics{figure_5} The lamina is released from rest with \(BC\) horizontal and below the level of the axis. Air resistance may be neglected, but a frictional couple opposes the motion. The couple has constant moment 44.1 N m about the axis. The angle through which the lamina has turned is denoted by \(\theta\) (see diagram).

1. Show that the angular acceleration is zero when \(\cos \theta = 0.25\). [3]
2. Hence find the maximum angular speed of the lamina. [5]