6.05f Vertical circle: motion including free fall

5. \begin{figure}[h] \end{figure} A uniform circular lamina has radius \(2 a\) and centre \(C\). The points \(P , Q , R\) and \(S\) on the lamina are the vertices of a square with centre \(C\) and \(C P = a\). Four circular discs, each of radius \(\frac { a } { 2 }\), with centres \(P , Q , R\) and \(S\), are removed from the lamina. The remaining lamina forms a template \(T\), as shown in Figure 1. The radius of gyration of \(T\) about an axis through \(C\), perpendicular to \(T\), is \(k\).

1. Show that \(k ^ { 2 } = \frac { 55 a ^ { 2 } } { 24 }\) The template \(T\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(T\) and passes through a point on its outer rim.
2. Write down an equation of rotational motion for \(T\) and deduce that the period of small oscillations of \(T\) about its stable equilibrium position is $$2 \pi \sqrt { } \left( \frac { 151 a } { 48 g } \right)$$

2. A uniform square lamina \(S\) has side \(2 a\). The radius of gyration of \(S\) about an axis through a vertex, perpendicular to \(S\), is \(k\).

1. Show that \(k ^ { 2 } = \frac { 8 a ^ { 2 } } { 3 }\). The lamina \(S\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(S\) and passes through a vertex.
2. By writing down an equation of rotational motion for \(S\), find the period of small oscillations of \(S\) about its position of stable equilibrium.

8. \begin{figure}[h] \end{figure} A uniform circular disc of radius \(2 a\) has centre \(O\). The points \(P , Q , R\) and \(S\) on the disc are the vertices of a square with centre \(O\) and \(O P = a\). Four circular holes, each of radius \(\frac { a } { 2 }\), and with centres \(P , Q , R\) and \(S\), are drilled in the disc to produce the lamina \(L\), shown shaded in Figure 1. The mass of \(L\) is \(M\).

1. Show that the moment of inertia of \(L\) about an axis through \(O\), and perpendicular to the plane of \(L\), is \(\frac { 55 M a ^ { 2 } } { 24 }\) The lamina \(L\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(L\) and which passes through a point \(A\) on the circumference of \(L\). At time \(t , A O\) makes an angle \(\theta\) with the downward vertical through \(A\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { 48 g } { 151 a } \sin \theta\)
3. Hence find the period of small oscillations of \(L\) about its position of stable equilibrium. The magnitude of the component, in a direction perpendicular to \(A O\), of the force exerted on \(L\) by the axis is \(X\).
4. Find \(X\) in terms of \(M , g\) and \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{57b98cdd-4121-4495-b500-185cbf3ff1a8-14_159_1662_2416_173}

7. A pendulum consists of a uniform circular disc, of radius \(a\) and mass \(4 m\), whose centre is fixed to the end \(B\) of a uniform \(\operatorname { rod } A B\). The rod has mass \(3 m\) and length \(4 l\), where \(2 l > a\). The rod lies in the same plane as the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the disc. The moment of inertia of the pendulum about \(L\) is \(2 m \left( a ^ { 2 } + 40 l ^ { 2 } \right)\).

1. Find the approximate period of small oscillations of the pendulum about its position of stable equilibrium. The pendulum is held with \(B\) vertically above \(A\) and is then slightly displaced from rest. In the subsequent motion the midpoint of \(A B\) strikes a small peg, which is fixed at the same horizontal level as \(A\), and the pendulum rebounds upwards. Immediately before it strikes the peg, the angular speed of the pendulum is \(\omega\).
2. Show that \(\omega ^ { 2 } = \frac { 22 g l } { \left( a ^ { 2 } + 40 l ^ { 2 } \right) }\) Immediately after it strikes the peg, the angular speed of the pendulum is \(\frac { 1 } { 2 } \omega\).
3. Find, in terms of \(m , g , a\) and \(l\), the magnitude of the impulse exerted on the peg by the pendulum.
4. Show that the size of the angle turned through by the pendulum, between it hitting the peg and it next coming to rest, is \(\arcsin \frac { 1 } { 4 }\).
   \includegraphics[max width=\textwidth, alt={}]{1242d28a-a4bd-4754-ac49-9b48de95b880-24_2632_1830_121_121}

6. (a) Show by integration that the moment of inertia of a uniform disc, of mass \(m\) and radius \(a\), about an axis through the centre of disc and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
(3 marks) \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(3 m\) and length \(2 a\). A uniform disc, of mass \(4 m\) and radius \(\frac { 1 } { 2 } a\), is attached to the rod with the centre of the disc lying on the rod a distance \(\frac { 3 } { 2 } a\) from \(A\). The rod lies in the plane of the disc, as shown in Fig. 1. The disc and rod together form a pendulum which is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the pendulum.
(b) Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 27 } { 2 } m a ^ { 2 }\). The pendulum makes small oscillations about its position of stable equilibrium.
(c) Show that the motion of the pendulum is approximately simple harmonic, and find the period of the oscillations.
(6 marks)

7. A uniform sphere, of mass \(m\) and radius \(a\), is free to rotate about a smooth fixed horizontal axis \(L\) which forms a tangent to the sphere. The sphere is hanging in equilibrium below the axis when it receives an impulse, causing it to rotate about \(L\) with an initial angular velocity of \(\sqrt { \frac { 18 g } { 7 a } }\). Show that, when the sphere has turned through an angle \(\theta\),

1. the angular speed \(\omega\) of the sphere is given by \(\omega ^ { 2 } = \frac { 2 g } { 7 a } ( 4 + 5 \cos \theta )\),
2. the angular acceleration of the sphere has magnitude \(\frac { 5 g } { 7 a } \sin \theta\).
3. Hence find the magnitude of the force exerted by the axis on the sphere when the sphere comes to instantaneous rest for the first time. END

6. The diagram shows a rollercoaster at an amusement park where a car is projected from a launch point \(O\) so that it performs a loop before instantaneously coming to rest at point \(C\). The car then performs the same journey in reverse. \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-5_677_1733_552_166} The loop section is modelled by considering the track to be a vertical circle of radius 10 m and the car as a particle of mass \(m\) kg moving on the inside surface of the circular loop. You may assume that the track is smooth. At point \(A\), which is the lowest point of the circle, the car has velocity \(u \mathrm {~ms} ^ { - 1 }\) such that \(u ^ { 2 } = 60 g\). When the car is at point \(B\) the radius makes an angle \(\theta\) with the downward vertical.

1. Find, in terms of \(\theta\) and \(g\), an expression for \(v ^ { 2 }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the car at \(B\).
2. Show that \(R \mathrm {~N}\), the reaction of the track on the car at \(B\), is given by $$R = m g ( 4 + 3 \cos \theta ) .$$
3. Explain why the expression for \(R\) in part (b) shows that the car will perform a complete loop.
4. This model predicts that the car will stop at \(C\) at a vertical height of 30 m above \(A\). However, after the car has completed the loop, the track becomes rough and the car only reaches a point \(D\) at a vertical height of 28 m above \(A\). The resistance to motion of the car beyond the loop is of constant magnitude \(\frac { m g } { 32 } \mathrm {~N}\). Calculate the length of the rough track between \(A\) and \(D\).

9 A light string, of natural length 0.5 m and modulus of elasticity 4 N , has one end attached to the ceiling of a room. A particle of mass 0.2 kg is attached to the free end of the string and hangs in equilibrium.

1. Find the extension of the string when the particle is in the equilibrium position. The particle is pulled down a further 0.5 m from the equilibrium position and released from rest. At time \(t\) seconds the displacement of the particle from the equilibrium position is \(x \mathrm {~m}\).
2. Show that, while the string is taut, the equation of motion is \(\ddot { x } = - 40 x\).
3. Find the time taken for the string to become slack for the first time.
4. Show that the particle comes to instantaneous rest 0.125 m below the ceiling.

10 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the free end of the string and the particle hangs at rest vertically below \(O\). The particle is projected horizontally with speed \(u\).

1. Find the tension in the string when it makes an angle \(\theta\) with the downward vertical, whilst the string remains taut.
2. Deduce that the particle will perform complete circles provided that \(u ^ { 2 } \geqslant 5 a g\).
3. It is given that \(u ^ { 2 } = 4 a g\). Find
   1. the tension in the string when \(\theta = 60 ^ { \circ }\),
   2. the value of \(\theta\), to the nearest degree, at the instant when the string becomes slack.

13 A fairground game consists of a small ball fixed to one end of a light inextensible string of length 0.6 m . The other end of the string is fixed to a point \(O\). A small bell is fixed 0.6 m vertically above \(O\). Initially the ball hangs vertically in equilibrium. The object of the game is to project the ball with an initial horizontal velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) so that it moves in a vertical circle and hits the bell.

1. Find the smallest possible value of \(u\) for which the ball hits the bell.
2. Given, instead, that the value of \(u\) is 5 , find the angle made by the string with the upward vertical at the moment when the string becomes slack.

\includegraphics{figure_5} 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 particle is moving in complete vertical circles with the string taut. When the particle is at the point \(P\), where \(OP\) makes an angle \(\alpha\) with the upward vertical through \(O\), its speed is \(u\). When the particle is at the point \(Q\), where angle \(QOP = 90°\), its speed is \(v\) (see diagram). It is given that \(\cos \alpha = \frac{4}{5}\).

1. Show that \(v^2 = u^2 + \frac{14}{5}ag\). [2]

The tension in the string when the particle is at \(Q\) is twice the tension in the string when the particle is at \(P\).

1. Obtain another equation relating \(u^2\), \(v^2\), \(a\) and \(g\), and hence find \(u\) in terms of \(a\) and \(g\). [5]
2. Find the least tension in the string during the motion. [3]

Answer only one of the following two alternatives. **EITHER** A particle \(P\), of mass \(m\), is able to move in a vertical circle on the smooth inner surface of a sphere with centre \(O\) and radius \(a\). Points \(A\) and \(B\) are on the inner surface of the sphere and \(AOB\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt{\left(\frac{21}{2}ag\right)}\) from \(A\) and begins to move in a vertical circle. At the lowest point of its path, vertically below \(O\), the particle \(P\) collides with a stationary particle \(Q\), of mass \(4m\), and rebounds. The speed acquired by \(Q\), as a result of the collision, is just sufficient for it to reach the point \(B\).

1. Find the speed of \(P\) and the speed of \(Q\) immediately after their collision. [7] In its subsequent motion, \(P\) loses contact with the inner surface of the sphere at the point \(D\), where the angle between \(OD\) and the upward vertical through \(O\) is \(\theta\).
2. Find \(\cos \theta\). [5] **OR** A farmer grows two different types of cherries, Type A and Type B. He assumes that the masses of each type are normally distributed. He chooses a random sample of 8 cherries of Type A. He finds that the sample mean mass is 15.1 g and that a 95% confidence interval for the population mean mass, \(\mu\) g, is \(13.5 \leqslant \mu \leqslant 16.7\).
3. Find an unbiased estimate for the population variance of the masses of cherries of Type A. [3] The farmer now chooses a random sample of 6 cherries of Type B and records their masses as follows. $$12.2 \quad 13.3 \quad 16.4 \quad 14.0 \quad 13.9 \quad 15.4$$
4. Test at the 5% significance level whether the mean mass of cherries of Type B is less than the mean mass of cherries of Type A. You should assume that the population variances for the two types of cherry are equal. [9]

A particle of mass \(m\) is attached to one end \(A\) of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and the particle hangs in equilibrium under gravity. The particle is projected horizontally so that it starts to move in a vertical circle. The string slackens after turning through an angle of \(120°\). Show that the speed of the particle is then \(\sqrt{\left(\frac{4}{3}ga\right)}\) and find the initial speed of projection. [5]

Answer only one of the following two alternatives. EITHER A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), is attached at one end to a fixed point and has a particle \(P\) of mass \(m\) attached to the other end. When \(P\) is hanging in equilibrium under gravity it is given a velocity \(\sqrt{(gl)}\) vertically downwards. At time \(t\) the downward displacement of \(P\) from its equilibrium position is \(x\). Show that, while the string is taut, $$\ddot{x} = -\frac{4g}{l}x.$$ [4] Find the speed of \(P\) when the length of the string is \(l\). [4] Show that the time taken for \(P\) to move from the lowest point to the highest point of its motion is $$\left(\frac{\pi}{3} + \frac{\sqrt{3}}{2}\right)\sqrt{\left(\frac{l}{g}\right)}.$$ [6] OR \includegraphics{figure_11} The scatter diagram shows a sample of size 5 of bivariate data, together with the regression line of \(y\) on \(x\). State what is minimised in obtaining this regression line, illustrating your answer on a copy of this diagram. [2] State, giving a reason, whether, for the data shown, the regression line of \(y\) on \(x\) is the same as the regression line of \(x\) on \(y\). [1] A car is travelling along a stretch of road with speed \(v\) km h\(^{-1}\) when the brakes are applied. The car comes to rest after travelling a further distance of \(z\) m. The values of \(z\) (and \(\sqrt{z}\)) for 8 different values of \(v\) are given in the table, correct to 2 decimal places.

\(v\)	25	30	35	40	45	50	55	60
\(z\)	2.83	4.63	4.84	5.29	9.73	10.30	14.82	15.21
\(\sqrt{z}\)	1.68	2.15	2.20	2.30	3.12	3.21	3.85	3.90

[\(\sum v = 340\), \(\sum v^2 = 15500\), \(\sum \sqrt{z} = 22.41\), \(\sum z = 67.65\), \(\sum v\sqrt{z} = 1022.15\).]

1. Calculate the product moment correlation coefficient between \(v\) and \(\sqrt{z}\). What does this indicate about the scatter diagram of the points \((v, \sqrt{z})\)? [4]
2. Given that the product moment correlation coefficient between \(v\) and \(z\) is 0.965, correct to 3 decimal places, state why the regression line of \(\sqrt{z}\) on \(v\) is more suitable than the regression line of \(z\) on \(v\), and find the equation of the regression line of \(\sqrt{z}\) on \(v\). [5]
3. Comment, in the context of the question, on the value of the constant term in the equation of the regression line of \(\sqrt{z}\) on \(v\). [2]

A particle \(P\) is describing simple harmonic motion of amplitude 5 m. Its speed is 6 m s\(^{-1}\) when it is 3 m from the centre of the motion. Find, in terms of \(\pi\), the period of the motion. [2] Find also

1. the maximum speed of \(P\), [2]
2. the magnitude of the maximum acceleration of \(P\). [2]

A uniform circular disc has diameter \(AB\), mass \(2m\) and radius \(a\). A particle of mass \(m\) is attached to the disc at \(B\). The disc is able to rotate about a smooth fixed horizontal axis through \(A\). The axis is tangential to the disc. Show that the moment of inertia of the system about the axis is \(\frac{5}{2}ma^2\). [4] The disc is held with \(AB\) horizontal and released. Find the angular speed of the system when \(B\) is directly below \(A\). [5] The disc is slightly displaced from the position of equilibrium in which \(B\) is below \(A\). At time \(t\) the angle between \(AB\) and the vertical is \(\theta\). Write down the equation of motion, and find the approximate period of small oscillations about the equilibrium position. [5]

Answer only one of the following two alternatives. **EITHER** A particle of mass 0.1 kg lies on a smooth horizontal table on the line between two points \(A\) and \(B\) on the table, which are 6 m apart. The particle is joined to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 60 N, and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 20 N. The mid-point of \(AB\) is \(M\), and \(O\) is the point between \(M\) and \(B\) at which the particle can rest in equilibrium. Show that \(MO = 0.2\) m. [4] The particle is held at \(M\) and then released. Show that the equation of motion is $$\frac{\mathrm{d}^2y}{\mathrm{d}t^2} = -500y,$$ where \(y\) metres is the displacement from \(O\) in the direction \(OB\) at time \(t\) seconds, and state the period of the motion. [5] For the instant when the particle is 0.3 m from \(M\) for the first time, find

1. the speed of the particle, [2]
2. the time taken, after release, to reach this position. [3]

**OR** The continuous random variable \(T\) has a negative exponential distribution with probability density function given by $$\mathrm{f}(t) = \begin{cases} \lambda\mathrm{e}^{-\lambda t} & t \geqslant 0, \\ 0 & \text{otherwise.} \end{cases}$$ Show that for \(t \geqslant 0\) the distribution function is given by F\((t) = 1 - \mathrm{e}^{-\lambda t}\). [2] The table below shows some values of F\((t)\) for the case when the mean is 20. Find the missing value. [2] It is thought that the lifetime of a species of insect under laboratory conditions has a negative exponential distribution with mean 20 hours. When observation starts there are 100 insects, which have been randomly selected. The lifetimes of the insects, in hours, are summarised in the table below. Calculate the expected values for each interval, assuming a negative exponential model with a mean of 20 hours, giving your values correct to 2 decimal places. [3] Perform a \(\chi^2\)-test of goodness of fit, at the 5% level of significance, in order to test whether a negative exponential distribution, with a mean of 20 hours, is a suitable model for the lifetime of this species of insect under laboratory conditions. [7]

The points \(A\) and \(B\) are on a smooth horizontal table at a distance \(8a\) apart. A particle \(P\) of mass \(m\) lies on the table on the line \(AB\), between \(A\) and \(B\). The particle is attached to \(A\) by a light elastic string of natural length \(3a\) and modulus of elasticity \(6mg\), and to \(B\) by a light elastic string of natural length \(2a\) and modulus of elasticity \(mg\). In equilibrium, \(P\) is at the point \(O\) on \(AB\).

1. Show that \(AO = 3.6a\). [4]

The particle is released from rest at the point \(C\) on \(AB\), between \(A\) and \(B\), where \(AC = 3.4a\).

1. Show that \(P\) moves in simple harmonic motion and state the period. [6]
2. Find the greatest speed of \(P\). [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 \(O\). 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]

An object is formed from a uniform circular disc, of radius \(2a\) and mass \(3M\), and a uniform rod \(AB\), of length \(4a\) and mass \(kM\), where \(k\) is a constant. The centre of the disc is \(O\). The end \(B\) of the rod is rigidly joined to a point on the circumference of the disc so that \(OBA\) is a straight line. The fixed horizontal axis \(l\) is in the plane of the object, passes through \(A\) and is perpendicular to \(AB\).

1. Show that the moment of inertia of the object about the axis \(l\) is \(3Ma^2(26 + k)\). [5]
2. The object is free to rotate about \(l\). Show that small oscillations of the object about \(l\) are approximately simple harmonic. Given that the period of these oscillations is \(4\pi\sqrt{\frac{a}{g}}\), find the value of \(k\). [6]

\includegraphics{figure_5} A thin uniform rod \(AB\) has mass \(\lambda M\) and length \(2a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(CBAO\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac{1}{2}a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).

1. Show that the moment of inertia of the object about \(L\) is \(\left(\frac{408 + 7\lambda}{12}\right)Ma^2\). [6]

The period of small oscillations of the object about \(L\) is \(5\pi\sqrt{\left(\frac{2a}{g}\right)}\).

1. Find the value of \(\lambda\). [6]