Questions — OCR M4 (100 questions)

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OCR M4 2004 June Q5
5 Two aircraft \(A\) and \(B\) are flying horizontally at the same height. \(A\) has constant velocity \(240 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(025 ^ { \circ }\), and \(B\) has constant velocity \(185 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(310 ^ { \circ }\).
  1. Find the magnitude and direction of the velocity of \(A\) relative to \(B\). Initially \(A\) is 4500 m due west of \(B\). For the instant during the subsequent motion when \(A\) and \(B\) are closest together, find
  2. the distance between \(A\) and \(B\),
  3. the bearing of \(A\) from \(B\).
OCR M4 2004 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{fb9e4e4a-953b-4e52-858e-438b4009e79c-3_428_595_221_806} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed horizontal axis through \(A\). A light elastic string has natural length \(a\) and modulus of elasticity \(m g\); one end is attached to \(B\) and the other end is attached to a light ring \(R\) which can slide along a smooth vertical wire. The wire is in the same vertical plane as \(A B\), and is at a distance \(a\) from \(A\). The rod \(A B\) makes an angle \(\theta\) with the upward vertical, where \(0 < \theta < \frac { 1 } { 2 } \pi\) (see diagram).
  1. Give a reason why the string \(R B\) is always horizontal.
  2. By considering potential energy, find the value of \(\theta\) for which the system is in equilibrium.
  3. Determine whether this position of equilibrium is stable or unstable.
OCR M4 2004 June Q7
7 A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The point \(P\) on the rod is such that \(A P = \frac { 2 } { 3 } a\).
  1. Prove by integration that the moment of inertia of the rod about an axis through \(P\) perpendicular to \(A B\) is \(\frac { 4 } { 9 } m a ^ { 2 }\). The axis through \(P\) is fixed and horizontal, and the rod can rotate without resistance in a vertical plane about this axis. The rod is released from rest in a horizontal position. Find, in terms of \(m\) and \(g\),
  2. the force acting on the rod at \(P\) immediately after the release of the rod,
  3. the force acting on the rod at \(P\) at an instant in the subsequent motion when \(B\) is vertically below \(P\).
OCR M4 2005 June Q1
1 A wheel is rotating freely with angular speed \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed axis through its centre. The moment of inertia of the wheel about the axis is \(0.65 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). A couple of constant moment is applied to the wheel, and in the next 5 seconds the wheel rotates through 180 radians.
  1. Find the angular acceleration of the wheel.
  2. Find the moment of the couple about the axis.
OCR M4 2005 June Q2
2 The region enclosed by the curve \(y = \sqrt { } x\) for \(0 \leqslant x \leqslant 9\), the \(x\)-axis and the line \(x = 9\) is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.
OCR M4 2005 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{b86c4b97-13a9-4aaf-8c95-20fe043b4532-2_653_406_727_857} A lamina has mass 1.5 kg . Two perpendicular lines \(A B\) and \(C D\) in the lamina intersect at the point \(X\). The centre of mass, \(G\), of the lamina lies on \(A B\), and \(X G = 0.2 \mathrm {~m}\) (see diagram). The moment of inertia of the lamina about \(A B\) is \(0.02 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), and the moment of inertia of the lamina about \(C D\) is \(0.12 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The lamina is free to rotate in a vertical plane about a fixed horizontal axis perpendicular to the lamina and passing through \(X\).
  1. The lamina makes small oscillations as a compound pendulum. Find the approximate period of these oscillations.
  2. The lamina starts at rest with \(G\) vertically below \(X\). A couple of constant moment 3.2 Nm about the axis is now applied to the lamina. Find the angular speed of the lamina when \(X G\) is first horizontal.
OCR M4 2005 June Q4
4 A boat \(A\) has constant velocity \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(110 ^ { \circ }\). A boat \(B\), which is initially 250 m due south of \(A\), moves with constant speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction which takes it as close as possible to \(A\).
  1. Find the bearing of the direction in which \(B\) moves.
  2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion.
OCR M4 2005 June Q5
5 In this question, \(a\) and \(k\) are positive constants.
The region enclosed by the curve \(y = a \mathrm { e } ^ { - \frac { x } { a } }\) for \(0 \leqslant x \leqslant k a\), the \(x\)-axis, the \(y\)-axis and the line \(x = k a\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of mass \(m\). Show that the moment of inertia of this solid about the \(x\)-axis is \(\frac { 1 } { 4 } m a ^ { 2 } \left( 1 + \mathrm { e } ^ { - 2 k } \right)\).
OCR M4 2005 June Q6
6 A uniform circular disc, of mass \(m\) and radius \(a\), has centre \(C\). The disc can rotate freely in a vertical plane about a fixed horizontal axis through the point \(A\) on the disc, where \(C A = \frac { 1 } { 2 } a\). The disc is released from rest in the position with \(C A\) horizontal. When the disc has rotated through an angle \(\theta\),
  1. show that the angular acceleration of the disc is \(\frac { 2 g \cos \theta } { 3 a }\),
  2. find the angular speed of the disc,
  3. find the components, parallel and perpendicular to \(C A\), of the force acting on the disc at the axis.
OCR M4 2005 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{b86c4b97-13a9-4aaf-8c95-20fe043b4532-3_585_801_991_647} A light rod \(A B\) of length \(2 a\) can rotate freely in a vertical plane about a fixed horizontal axis through \(A\). A particle of mass \(m\) is attached to the rod at \(B\). A fixed smooth ring \(R\) lies in the same vertical plane as the rod, where \(A R = a\) and \(A R\) makes an angle \(\frac { 1 } { 4 } \pi\) above the horizontal. A light elastic string, of natural length \(a\) and modulus of elasticity \(m g \sqrt { } 2\), passes through the ring \(R\); one end is fixed to \(A\) and the other end is fixed to \(B\). The rod makes an angle \(\theta\) below the horizontal, where \(- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi\) (see diagram).
  1. Use the cosine rule to show that \(R B ^ { 2 } = a ^ { 2 } ( 5 - ( 2 \sqrt { } 2 ) \cos \theta + ( 2 \sqrt { } 2 ) \sin \theta )\).
  2. Show that \(\theta = 0\) is a position of stable equilibrium.
  3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - k \sin \theta\), expressing the constant \(k\) in terms of \(a\) and \(g\), and hence write down the approximate period of small oscillations about the equilibrium position \(\theta = 0\).
OCR M4 2006 June Q1
1 A straight rod \(A B\) of length \(a\) has variable density. At a distance \(x\) from \(A\) its mass per unit length is \(k ( a + 2 x )\), where \(k\) is a positive constant. Find the distance from \(A\) of the centre of mass of the rod.
OCR M4 2006 June Q2
2 A flywheel takes the form of a uniform disc of mass 8 kg and radius 0.15 m . It rotates freely about an axis passing through its centre and perpendicular to the disc. A couple of constant moment is applied to the flywheel. The flywheel turns through an angle of 75 radians while its angular speed increases from \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Find the moment of the couple about the axis. When the flywheel is rotating with angular speed \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\), it locks together with a second flywheel which is mounted on the same axis and is at rest. Immediately afterwards, both flywheels rotate together with the same angular speed \(9 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  2. Find the moment of inertia of the second flywheel about the axis.
OCR M4 2006 June Q3
3 The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\) and the curve \(y = \frac { 1 } { x ^ { 2 } }\) for \(1 \leqslant x \leqslant 2\), is occupied by a uniform lamina of mass 24 kg . The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-2_623_601_1409_706} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point at \(A\). A particle of mass \(2 m\) is attached to the rod at \(B\). A light elastic string, with natural length \(a\) and modulus of elasticity \(5 m g\), passes through a fixed smooth ring \(R\). One end of the string is fixed to \(A\) and the other end is fixed to the mid-point \(C\) of \(A B\). The ring \(R\) is at the same horizontal level as \(A\), and is at a distance \(a\) from \(A\). The rod \(A B\) and the ring \(R\) are in a vertical plane, and \(R C\) is at an angle \(\theta\) above the horizontal, where \(0 < \theta < \frac { 1 } { 4 } \pi\), so that the acute angle between \(A B\) and the horizontal is \(2 \theta\) (see diagram).
  1. By considering the energy of the system, find the value of \(\theta\) for which the system is in equilibrium.
  2. Determine whether this position of equilibrium is stable or unstable.
OCR M4 2006 June Q5
5 A uniform rectangular lamina \(A B C D\) has mass 20 kg and sides of lengths \(A B = 0.6 \mathrm {~m}\) and \(B C = 1.8 \mathrm {~m}\). 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 \(A B\).
  1. Show that the moment of inertia of the lamina about the axis is \(22.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-3_442_541_477_800} The lamina is released from rest with \(B C\) 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 Nm about the axis. The angle through which the lamina has turned is denoted by \(\theta\) (see diagram).
  2. Show that the angular acceleration is zero when \(\cos \theta = 0.25\).
  3. Hence find the maximum angular speed of the lamina.
    \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-3_633_838_1356_650} A ship \(P\) is moving with constant velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(110 ^ { \circ }\). A second ship \(Q\) is moving with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line. At one instant \(Q\) is at the point \(X\), and \(P\) is 7400 m from \(Q\) on a bearing of \(050 ^ { \circ }\) (see diagram). In the subsequent motion, the shortest distance between \(P\) and \(Q\) is 1790 m .
  4. Show that one possible direction for the velocity of \(Q\) relative to \(P\) has bearing \(036 ^ { \circ }\), to the nearest degree, and find the bearing of the other possible direction of this relative velocity. Given that the velocity of \(Q\) relative to \(P\) has bearing \(036 ^ { \circ }\), find
  5. the bearing of the direction in which \(Q\) is moving,
  6. the magnitude of the velocity of \(Q\) relative to \(P\),
  7. the time taken for \(Q\) to travel from \(X\) to the position where the two ships are closest together,
  8. the bearing of \(P\) from \(Q\) when the two ships are closest together.
    \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-4_560_1180_265_467} A uniform rod \(A B\) has mass \(m\) and length \(6 a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(C\) on the rod, where \(A C = a\). The angle between \(A B\) and the upward vertical is \(\theta\), and the force acting on the rod at \(C\) has components \(R\) parallel to \(A B\) and \(S\) perpendicular to \(A B\) (see diagram). The rod is released from rest in the position where \(\theta = \frac { 1 } { 3 } \pi\). Air resistance may be neglected.
  9. Find the angular acceleration of the rod in terms of \(a , g\) and \(\theta\).
  10. Show that the angular speed of the rod is \(\sqrt { \frac { 2 g ( 1 - 2 \cos \theta ) } { 7 a } }\).
  11. Find \(R\) and \(S\) in terms of \(m , g\) and \(\theta\).
  12. When \(\cos \theta = \frac { 1 } { 3 }\), show that the force acting on the rod at \(C\) is vertical, and find its magnitude.
OCR M4 2007 June Q1
1 The driveshaft of an electric motor begins to rotate from rest and has constant angular acceleration. In the first 8 seconds it turns through 56 radians.
  1. Find the angular acceleration.
  2. Find the angle through which the driveshaft turns while its angular speed increases from \(20 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(36 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
OCR M4 2007 June Q2
2 The region \(R\) is bounded by the curve \(y = \sqrt { 4 a ^ { 2 } - x ^ { 2 } }\) for \(0 \leqslant x \leqslant a\), the \(x\)-axis, the \(y\)-axis and the line \(x = a\), where \(a\) is a positive constant. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.
OCR M4 2007 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-2_392_746_908_645} A non-uniform rectangular lamina \(A B C D\) has mass 6 kg . The centre of mass \(G\) of the lamina is 0.8 m from the side \(A D\) and 0.5 m from the side \(A B\) (see diagram). The moment of inertia of the lamina about \(A D\) is \(6.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\) and the moment of inertia of the lamina about \(A B\) is \(2.8 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The lamina rotates in a vertical plane about a fixed horizontal axis which passes through \(A\) and is perpendicular to the lamina.
  1. Write down the moment of inertia of the lamina about this axis. The lamina is released from rest in the position where \(A B\) and \(D C\) are horizontal and \(D C\) is above \(A B\). A frictional couple of constant moment opposes the motion. When \(A B\) is first vertical, the angular speed of the lamina is \(2.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  2. Find the moment of the frictional couple.
  3. Find the angular acceleration of the lamina immediately after it is released.
OCR M4 2007 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-3_698_505_275_801} A uniform solid cylinder has radius \(a\), height \(3 a\), and mass \(M\). The line \(A B\) is a diameter of one of the end faces of the cylinder (see diagram).
  1. Show by integration that the moment of inertia of the cylinder about \(A B\) is \(\frac { 13 } { 4 } M a ^ { 2 }\). (You may assume that the moment of inertia of a uniform disc of mass \(m\) and radius \(a\) about a diameter is \(\frac { 1 } { 4 } m a ^ { 2 }\).) The line \(A B\) is now fixed in a horizontal position and the cylinder rotates freely about \(A B\), making small oscillations as a compound pendulum.
  2. Find the approximate period of these small oscillations, in terms of \(a\) and \(g\).
OCR M4 2007 June Q5
5 A ship \(S\) is travelling with constant speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(345 ^ { \circ }\). A patrol boat \(B\) spots the ship \(S\) when \(S\) is 2400 m from \(B\) on a bearing of \(050 ^ { \circ }\). The boat \(B\) sets off in pursuit, travelling with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line.
  1. Given that \(v = 16\), find the bearing of the course which \(B\) should take in order to intercept \(S\), and the time taken to make the interception.
  2. Given instead that \(v = 10\), find the bearing of the course which \(B\) should take in order to get as close as possible to \(S\).
    \includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-4_337_954_278_544} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The point \(P\) on the rod is such that \(A P = \frac { 2 } { 3 } a\). The rod is placed in a horizontal position perpendicular to the edge of a rough horizontal table, with \(A P\) in contact with the table and \(P B\) overhanging the edge. The rod is released from rest in this position. When it has rotated through an angle \(\theta\), and no slipping has occurred at \(P\), the normal reaction acting on the rod at \(P\) is \(R\) and the frictional force is \(F\) (see diagram).
  3. Show that the angular acceleration of the rod is \(\frac { 3 g \cos \theta } { 4 a }\).
  4. Find the angular speed of the rod, in terms of \(a , g\) and \(\theta\).
  5. Find \(F\) and \(R\) in terms of \(m , g\) and \(\theta\).
  6. Given that the coefficient of friction between the rod and the edge of the table is \(\mu\), show that the rod is on the point of slipping at \(P\) when \(\tan \theta = \frac { 1 } { 2 } \mu\).
    \includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-5_677_624_269_753} A smooth circular wire, with centre \(O\) and radius \(a\), is fixed in a vertical plane. The highest point on the wire is \(A\) and the lowest point on the wire is \(B\). A small ring \(R\) of mass \(m\) moves freely along the wire. A light elastic string, with natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\), 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, so that \(O R\) makes an angle \(2 \theta\) with the downward vertical (see diagram). You may assume that the string does not become slack.
  7. Taking \(A\) as the level for zero gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = m g a \left( \frac { 1 } { 4 } - \cos \theta - \cos ^ { 2 } \theta \right) .$$
  8. Show that \(\theta = 0\) is the only position of equilibrium.
  9. By differentiating the energy equation with respect to time \(t\), show that $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { g } { 4 a } \sin \theta ( 1 + 2 \cos \theta ) .$$
  10. Deduce the approximate period of small oscillations about the equilibrium position \(\theta = 0\).
OCR M4 2008 June Q1
1 Two flywheels \(F\) and \(G\) are rotating freely, about the same axis and in the same direction, with angular speeds \(21 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(36 \mathrm { rad } \mathrm { s } ^ { - 1 }\) respectively. The flywheels come into contact briefly, and immediately afterwards the angular speeds of \(F\) and \(G\) are \(28 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(34 \mathrm { rad } \mathrm { s } ^ { - 1 }\), respectively, in the same direction. Given that the moment of inertia of \(F\) about the axis is \(1.5 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), find the moment of inertia of \(G\) about the axis.
OCR M4 2008 June Q2
2 A rotating turntable is slowing down with constant angular deceleration. It makes 16 revolutions as its angular speed decreases from \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to rest.
  1. Find the angular deceleration of the turntable.
  2. Find the angular speed of the turntable at the start of its last complete revolution before coming to rest.
  3. Find the time taken for the turntable to make its last complete revolution before coming to rest.
OCR M4 2008 June Q3
3 The region bounded by the curve \(y = 2 x + x ^ { 2 }\) for \(0 \leqslant x \leqslant 3\), the \(x\)-axis, and the line \(x = 3\), is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.
OCR M4 2008 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-2_823_650_1318_751} A boat \(A\) is travelling with constant speed \(6.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(075 ^ { \circ }\). Boat \(B\) is travelling with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(025 ^ { \circ }\). At one instant, \(A\) is 2500 m due north of \(B\) (see diagram).
  1. Find the magnitude and bearing of the velocity of \(A\) relative to \(B\).
  2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion.
OCR M4 2008 June Q5
5 The region bounded by the curve \(y = \sqrt { a x }\) for \(a \leqslant x \leqslant 4 a\) (where \(a\) is a positive constant), the \(x\)-axis, and the lines \(x = a\) and \(x = 4 a\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution of mass \(m\).
  1. Show that the moment of inertia of this solid about the \(x\)-axis is \(\frac { 7 } { 5 } m a ^ { 2 }\). The solid is free to rotate about a fixed horizontal axis along the line \(y = a\), and makes small oscillations as a compound pendulum.
  2. Find, in terms of \(a\) and \(g\), the approximate period of these small oscillations.
    \includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-3_734_862_813_644} A uniform rectangular lamina \(A B C D\) has mass \(m\) and sides \(A B = 2 a\) and \(B C = 3 a\). The mid-point of \(A B\) is \(P\) and the mid-point of \(C D\) is \(Q\). The lamina is rotating freely in a vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the point \(X\) on \(P Q\) where \(P X = a\). Air resistance may be neglected. When \(Q\) is vertically above \(X\), the angular speed is \(\sqrt { \frac { 9 g } { 10 a } }\). When \(X Q\) makes an angle \(\theta\) with the upward vertical, the angular speed is \(\omega\), and the force acting on the lamina at \(X\) has components \(R\) parallel to \(P Q\) and \(S\) parallel to \(B A\) (see diagram).
OCR M4 2008 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-4_622_767_269_689} Particles \(P\) and \(Q\), with masses \(3 m\) and \(2 m\) respectively, are connected by a light inextensible string passing over a smooth light pulley. The particle \(P\) is connected to the floor by a light spring \(S _ { 1 }\) with natural length \(a\) and modulus of elasticity mg . The particle \(Q\) is connected to the floor by a light spring \(S _ { 2 }\) with natural length \(a\) and modulus of elasticity \(2 m g\). The sections of the string not in contact with the pulley, and the two springs, are vertical. Air resistance may be neglected. The particles \(P\) and \(Q\) move vertically and the string remains taut; when the length of \(S _ { 1 }\) is \(x\), the length of \(S _ { 2 }\) is ( \(3 a - x\) ) (see diagram).
  1. Find the total potential energy of the system (taking the floor as the reference level for gravitational potential energy). Hence show that \(x = \frac { 4 } { 3 } a\) is a position of stable equilibrium.
  2. By differentiating the energy equation, and substituting \(x = \frac { 4 } { 3 } a + y\), show that the motion is simple harmonic, and find the period.