6.02h Elastic PE: 1/2 k x^2

4 A particle \(P\) of mass 0.2 kg is suspended from a fixed point \(O\) by a light elastic string of natural length 0.7 m and modulus of elasticity \(3.5 \mathrm {~N} . P\) is at the equilibrium position when it is projected vertically downwards with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after being set in motion \(P\) is \(x \mathrm {~m}\) below the equilibrium position and has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the equilibrium position of \(P\) is 1.092 m below \(O\).
2. Prove that \(P\) moves with simple harmonic motion, and calculate the amplitude.
3. Calculate \(x\) and \(v\) when \(t = 0.4\).

7 \includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-4_721_691_269_726} The diagram shows a particle \(P\) of mass 0.5 kg attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius 1.2 m . The string has natural length 0.6 m and modulus of elasticity \(6.86 \mathrm {~N} . P\) is released from rest at a point on the surface of the sphere where the acute angle \(A O P\) is at least 0.5 radians.

1. (a) For the case angle \(A O P = \alpha , P\) remains at rest. Show that \(\sin \alpha = 2.8 \alpha - 1.4\).
   (b) Use the iterative formula $$\alpha _ { n + 1 } = \frac { \sin \alpha _ { n } } { 2.8 } + 0.5 ,$$ with \(\alpha _ { 1 } = 0.8\), to find \(\alpha\) correct to 2 significant figures.
2. Given instead that angle \(A O P = 0.5\) radians when \(P\) is released, find the speed of \(P\) when angle \(A O P = 0.8\) radians, given that \(P\) is at all times in contact with the surface of the sphere. State whether the speed of \(P\) is increasing or decreasing when angle \(A O P = 0.8\) radians.

5 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 0.75 mgN . The other end of the string is attached to a fixed point \(O\) of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) is released from rest at \(O\) and moves down the plane.

1. Show that the maximum speed of \(P\) is reached when the extension of the string is 0.8 m .
2. Find the maximum speed of \(P\).
3. Find the maximum displacement of \(P\) from \(O\).

7 A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to one end of a light elastic string of natural length 3.2 m and modulus of elasticity \(4 m g \mathrm {~N}\). The other end of the string is attached to a fixed point \(A\). The particle is released from rest at a point 4.8 m vertically below \(A\). At time \(t \mathrm {~s}\) after \(P\) 's release \(P\) is ( \(4 + x ) \mathrm { m }\) below \(A\).

1. Show that \(4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 49 x\). \(P\) 's motion is simple harmonic.
2. Write down the amplitude of \(P\) 's motion and show that the string becomes slack instantaneously at intervals of approximately 1.8 s . A particle \(Q\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(B\). The particle is released from rest with the string taut and inclined at a small angle with the downward vertical. At time \(t \mathrm {~s}\) after \(Q\) 's release \(B Q\) makes an angle of \(\theta\) radians with the downward vertical.
3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } \approx - \frac { g } { L } \theta\). The period of the simple harmonic motion to which \(Q\) 's motion approximates is the same as the period of \(P\) 's motion.
4. Given that \(\theta = 0.08\) when \(t = 0\), find the speed of \(Q\) when \(t = 0.25\).

3 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-3_387_181_274_982} \(A\) and \(B\) are fixed points with \(B\) at a distance of 1.8 m vertically below \(A\). One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to \(A\), and one end of an identical elastic string is attached to \(B\). A particle \(P\) of weight 12 N is attached to the other ends of the strings (see diagram).

1. Verify that \(P\) is in equilibrium when it is at a distance of 1.05 m vertically below \(A\). \(P\) is released from rest at the point 1.2 m vertically below \(A\) and begins to move.
2. Show that, when \(P\) is \(x \mathrm {~m}\) below its equilibrium position, the tensions in \(P A\) and \(P B\) are \(( 18 + 40 x ) \mathrm { N }\) and \(( 6 - 40 x ) \mathrm { N }\) respectively.
3. Show that \(P\) moves with simple harmonic motion of period 0.777 s , correct to 3 significant figures.
4. Find the speed with which \(P\) passes through the equilibrium position. \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-3_540_655_1564_744} 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. With the string taut and horizontal, \(P\) is projected with a velocity of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downward. \(P\) begins to move in a vertical circle with centre \(O\). While the string remains taut the angular displacement of \(O P\) is \(\theta\) radians from its initial position, and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).

7 A particle of mass 0.8 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to a fixed point \(O\). The particle is held at rest at \(O\) and then released. When the extension of the string is \(x \mathrm {~m}\), the particle is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy show that \(v ^ { 2 } = 39.2 + 19.6 x - 12.5 x ^ { 2 }\).
2. Hence find
   1. the maximum extension of the string,
   2. the maximum speed of the particle,
   3. the maximum magnitude of the acceleration of the particle.

5 A light elastic string of natural length 1.6 m has modulus of elasticity 120 N . One end of the string is attached to a fixed point \(O\) and the other end is attached to a particle \(P\) of weight 1.5 N . The particle is released from rest at the point \(A\), which is 2.1 m vertically below \(O\). It comes instantaneously to rest at \(B\), which is vertically above \(O\).

1. Verify that the distance \(A B\) is 4 m .
2. Find the maximum speed of \(P\) during its upward motion from \(A\) to \(B\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_351_442_303_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_394_648_260_1018} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A light inextensible string of length \(0.8 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.4 kg and 0.58 kg respectively, attached to its ends. The string passes over a smooth horizontal cylinder of radius 0.8 m , which is fixed with its axis horizontal and passing through a fixed point \(O\). The string is held at rest in a vertical plane perpendicular to the axis of the cylinder, with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the cylinder 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).

7 A particle \(P\) of mass 0.5 kg is attached to one end of each of two identical light elastic strings of natural length 1.6 m and modulus of elasticity 19.6 N . The other ends of the strings are attached to fixed points \(A\) and \(B\) on a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is 4.8 m and \(A\) is higher than \(B\).

1. Find the distance \(A P\) for which \(P\) is in equilibrium on the line \(A B\). \(P\) is released from rest at a point on \(A B\) where both strings are taut. The strings remain taut during the subsequent motion of \(P\) and \(t\) seconds after release the distance \(A P\) is \(( 2.5 + x ) \mathrm { m }\).
2. Use Newton's second law to obtain an equation of the form \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = k x\). State the property of the constant \(k\) for which the equation indicates that \(P\) 's motion is simple harmonic, and find the period of this motion.
3. Given that \(x = 0.5\) when \(t = 0\), find the values of \(x\) for which the speed of \(P\) is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). www.ocr.org.uk after the live examination series.
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5 A particle \(P\) of mass 0.05 kg is suspended from a fixed point \(O\) by a light elastic string of natural length 0.5 m and modulus of elasticity 2.45 N .

1. Show that the equilibrium position of \(P\) is 0.6 m below \(O\). \(P\) is held at rest at a point 0.675 m vertically below \(O\) and then released. At time \(t \mathrm {~s}\) after \(P\) is released, its downward displacement from the equilibrium position is \(x \mathrm {~m}\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 98 x\).
3. Find the value of \(x\) and the magnitude and direction of the velocity of \(P\) when \(t = 0.2\).

6 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-4_638_473_260_836} A particle \(P\), of mass 3.5 kg , is in equilibrium suspended from the top \(A\) of a smooth slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 40 } { 49 }\), by an elastic rope of natural length 4 m and modulus of elasticity 112 N (see diagram). Another particle \(Q\), of mass 0.5 kg , is released from rest at \(A\) and slides freely downwards until it reaches \(P\) and becomes attached to it.

1. Find the value of \(V ^ { 2 }\), where \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(Q\) immediately before it becomes attached to \(P\), and show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(\frac { 1 } { 2 } \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The combined particles slide downwards for a distance of \(X \mathrm {~m}\), before coming instantaneously to rest at \(B\).
2. Show that \(28 X ^ { 2 } - 8 X - 5 = 0\).

4 One end of a light elastic string, of natural length 0.75 m and modulus of elasticity 44.1 N , is attached to a fixed point \(O\). A particle \(P\) of mass 1.8 kg is attached to the other end of the string. \(P\) is released from rest at \(O\) and falls vertically. Assuming there is no air resistance, find

1. the extension of the string when \(P\) is at its lowest position,
2. the acceleration of \(P\) at its lowest position.

7 \includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-4_351_314_255_861} One end of a light elastic string, of natural length \(\frac { 2 } { 3 } R \mathrm {~m}\) and with modulus of elasticity 1.2 mgN , is attached to the highest point \(A\) of a smooth fixed sphere with centre \(O\) and radius \(R \mathrm {~m}\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string and is in contact with the surface of the sphere, where the angle \(A O P\) is equal to \(\theta\) radians (see diagram).

1. Given that \(P\) is in equilibrium at the point where \(\theta = \alpha\), show that \(1.8 \alpha - \sin \alpha - 1.2 = 0\). Hence show that \(\alpha = 1.18\) correct to 3 significant figures. \(P\) is now released from rest at the point of the surface of the sphere where \(\theta = \frac { 2 } { 3 }\), and starts to move downwards on the surface. For an instant when \(\theta = \alpha\),
2. state the direction of the acceleration of \(P\),
3. find the magnitude of the acceleration of \(P\).

5 A particle \(P\), of mass 2.5 kg , is in equilibrium suspended from a fixed point \(A\) by a light elastic string of natural length 3 m and modulus of elasticity 36.75 N . Another particle \(Q\), of mass 1 kg , is released from rest at \(A\) and falls freely until it reaches \(P\) and becomes attached to it.

1. Show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(2 \sqrt { 2 } \mathrm {~ms} ^ { - 1 }\). The combined particles fall a further distance \(X \mathrm {~m}\) before coming to instantaneous rest.
2. Find a quadratic equation satisfied by \(X\), and show that it simplifies to \(35 X ^ { 2 } - 56 X - 80 = 0\).

6 A bungee jumper of mass 70 kg is joined to a fixed point \(O\) by a light elastic rope of natural length 30 m and modulus of elasticity 1470 N . The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.

1. Find the distance fallen by the jumper when maximum speed is reached.
2. Show that this maximum speed is \(26.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. Find the extension of the rope when the jumper is at the lowest position. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_543_616_310_301} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_668_709_267_1135} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A smooth horizontal cylinder of radius 0.6 m is fixed with its axis horizontal and passing through a fixed point \(O\). A light inextensible string of length \(0.6 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.3 kg and 0.4 kg respectively, attached at its ends. The string passes over the cylinder and is held at rest with \(P , O\) and \(Q\) in a straight horizontal line (see Fig. 1). The string is released and \(Q\) begins to descend. When the line \(O P\) makes an angle \(\theta\) radians, \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), with the horizontal, the particles have speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).

1. By considering the total energy of the system, or otherwise, show that $$v ^ { 2 } = 6.72 \theta - 5.04 \sin \theta .$$
2. Show that the magnitude of the contact force between \(P\) and the cylinder is $$( 5.46 \sin \theta - 3.36 \theta ) \text { newtons. }$$ Hence find the value of \(\theta\) for which the magnitude of the contact force is greatest.
3. Find the transverse component of the acceleration of \(P\) in terms of \(\theta\).

5 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_549_447_253_849} Two uniform rods \(A B\) and \(B C\), each of length 1.4 m and weight 80 N , are freely jointed to each other at \(B\), and \(A B\) is freely jointed to a fixed point at \(A\). They are held in equilibrium with \(A B\) at an angle \(\alpha\) to the horizontal, and \(B C\) at an angle of \(60 ^ { \circ }\) to the horizontal, by a light string, perpendicular to \(B C\), attached to \(C\) (see diagram).

1. By taking moments about \(B\) for \(B C\), calculate the tension in the string. Hence find the horizontal and vertical components of the force acting on \(B C\) at \(B\).
2. Find \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_691_665_1370_740} A circus performer \(P\) of mass 80 kg is suspended from a fixed point \(O\) by an elastic rope of natural length 5.25 m and modulus of elasticity \(2058 \mathrm {~N} . P\) is in equilibrium at a point 5 m above a safety net. A second performer \(Q\), also of mass 80 kg , falls freely under gravity from a point above \(P\). \(P\) catches \(Q\) and together they begin to descend vertically with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) (see diagram). The performers are modelled as particles.

1 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 1.8 m and modulus of elasticity 1.35 mg N . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. \(P\) is held at rest at a point on the surface 3 m from \(O\). The particle is then released. Find

1. the initial acceleration of \(P\),
2. the speed of \(P\) at the instant the string becomes slack.

5 One end of a light elastic string, of natural length 0.78 m and modulus of elasticity 0.8 mgN , is attached to a fixed point \(O\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. \(P\) is released from rest at \(O\) and moves down the plane without reaching the bottom. Find

1. the maximum speed of \(P\) in the subsequent motion,
2. the distance of \(P\) from \(O\) when it is at its lowest point.

1 \includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-2_435_665_255_699} A small object \(W\) of weight 100 N is attached to one end of each of two parallel light elastic strings. One string is of natural length 0.4 m and has modulus of elasticity 20 N ; the other string is of natural length 0.6 m and has modulus of elasticity 30 N . The upper ends of both strings are attached to a horizontal ceiling and \(W\) hangs in equilibrium at a distance \(d \mathrm {~m}\) below the ceiling (see diagram). Find \(d\).

7 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 39.2 mN . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).

1. Show that, while the string is in tension, the particle performs simple harmonic motion about a point 1 m below \(O\).
2. Show that when \(P\) is at its lowest point the extension of the string is 0.8 m .
3. Find the time after its release that \(P\) first reaches its lowest point.
4. Find the velocity of \(P 0.8 \mathrm {~s}\) after it is released from \(O\). }{www.ocr.org.uk}) after the live examination series.
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2 One end of a light elastic string, of natural length 0.6 m and modulus of elasticity 30 N , is attached to a fixed point \(O\). A particle \(P\) of weight 48 N is attached to the other end of the string. \(P\) is released from rest at a point \(d \mathrm {~m}\) vertically below \(O\). Subsequently \(P\) just reaches \(O\).

1. Find \(d\).
2. Find the magnitude and direction of the acceleration of \(P\) when it has travelled 1.3 m from its point of release.

7 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_382_773_1567_648} One end of a light elastic string, of natural length 0.3 m , is attached to a fixed point \(O\) on a smooth plane that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.2\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. The string lies along a line of greatest slope of the plane and has modulus of elasticity \(2.45 m \mathrm {~N}\) (see diagram).

1. Show that in the equilibrium position the extension of the string is 0.24 m . \(P\) is given a velocity of \(0.3 \mathrm {~ms} ^ { - 1 }\) down the plane from the equilibrium position.
2. Show that \(P\) performs simple harmonic motion with period 2.20 s (correct to 3 significant figures), and find the amplitude of the motion.
3. Find the distance of \(P\) from \(O\) and the velocity of \(P\) at the instant 1.5 seconds after \(P\) is set in motion.

4 For a bungee jump, a girl is joined to a fixed point \(O\) of a bridge by an elastic rope of natural length 25 m and modulus of elasticity 1320 N . The girl starts from rest at \(O\) and falls vertically. The lowest point reached by the girl is 60 m vertically below \(O\). The girl is modelled as a particle, the rope is assumed to be light, and air resistance is neglected.

1. Find the greatest tension in the rope during the girl's jump.
2. Use energy considerations to find
   1. the mass of the girl,
   2. the speed of the girl when she has fallen half way to the lowest point.

3 Two fixed points X and Y are 14.4 m apart and XY is horizontal. The midpoint of XY is M . A particle P is connected to X and to Y by two light elastic strings. Each string has natural length 6.4 m and modulus of elasticity 728 N . The particle P is in equilibrium when it is 3 m vertically below M, as shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Find the tension in each string when P is in the equilibrium position.
2. Show that the mass of P is 12.5 kg . The particle P is released from rest at M , and moves in a vertical line.
3. Find the acceleration of P when it is 2.1 m vertically below M .
4. Explain why the maximum speed of P occurs at the equilibrium position.
5. Find the maximum speed of P .

1

1. 1. Find the dimensions of power. In a particle accelerator operating at power \(P\), a charged sphere of radius \(r\) and density \(\rho\) has its speed increased from \(u\) to \(2 u\) over a distance \(x\). A student derives the formula $$x = \frac { 28 \pi r ^ { 3 } u ^ { 2 } \rho } { 9 P }$$
   2. Show that this formula is not dimensionally consistent.
   3. Given that there is only one error in this formula for \(x\), obtain the correct formula.
2. A light elastic string, with natural length 1.6 m and stiffness \(150 \mathrm { Nm } ^ { - 1 }\), is stretched between fixed points A and B which are 2.4 m apart on a smooth horizontal surface.
   1. Find the energy stored in the string. A particle is attached to the mid-point of the string. The particle is given a horizontal velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to AB (see Fig. 1.1), and it comes instantaneously to rest after travelling a distance of 0.9 m (see Fig. 1.2). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-2_524_305_1274_639} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-2_524_305_1274_1128} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   2. Find the mass of the particle.

3 A fixed point A is 12 m vertically above a fixed point B. A light elastic string, with natural length 3 m and modulus of elasticity 1323 N , has one end attached to A and the other end attached to a particle P of mass 15 kg . Another light elastic string, with natural length 4.5 m and modulus of elasticity 1323 N , has one end attached to B and the other end attached to P .

1. Verify that, in the equilibrium position, \(\mathrm { AP } = 5 \mathrm {~m}\). The particle P now moves vertically, with both strings AP and BP remaining taut throughout the motion. The displacement of P above the equilibrium position is denoted by \(x \mathrm {~m}\) (see Fig. 3). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-4_405_360_751_849} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Show that the tension in the string AP is \(441 ( 2 - x ) \mathrm { N }\) and find the tension in the string BP .
3. Show that the motion of P is simple harmonic, and state the period. The minimum length of AP during the motion is 3.5 m .
4. Find the maximum length of AP .
5. Find the speed of P when \(\mathrm { AP } = 4.1 \mathrm {~m}\).
6. Find the time taken for AP to increase from 3.5 m to 4.5 m .