Prove SHM and find period

5
\includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_576_535_258_804} A particle \(P\) of mass 0.3 kg is moving in a vertical circle. It is attached to the fixed point \(O\) at the centre of the circle by a light inextensible string of length 1.5 m . When the string makes an angle of \(40 ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Air resistance may be neglected.

1. Find the radial and transverse components of the acceleration of \(P\) at this instant. In the subsequent motion, with the string still taut and making an angle \(\theta ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Use conservation of energy to show that \(v ^ { 2 } \approx 19.7 + 29.4 \cos \theta ^ { \circ }\).
3. Find the tension in the string in terms of \(\theta\).
4. Find the value of \(v\) at the instant when the string becomes slack.
   \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_574_842_1640_664} A step-ladder is modelled as two uniform rods \(A B\) and \(A C\), freely jointed at \(A\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) in contact with a rough horizontal surface. The rods have equal lengths; \(A B\) has weight 150 N and \(A C\) has weight 270 N . The point \(A\) is 2.5 m vertically above the surface, and \(B C = 1.6 \mathrm {~m}\) (see diagram).
5. Find the horizontal and vertical components of the force acting on \(A C\) at \(A\).
6. The coefficient of friction has the same value \(\mu\) at \(B\) and at \(C\), and the step-ladder is on the point of slipping. Giving a reason, state whether the equilibrium is limiting at \(B\) or at \(C\), and find \(\mu\).
   \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-4_648_227_269_982} Two points \(A\) and \(B\) lie on a vertical line with \(A\) at a distance 2.6 m above \(B\). A particle \(P\) of mass 10 kg is joined to \(A\) by an elastic string and to \(B\) by another elastic string (see diagram). Each string has natural length 0.8 m and modulus of elasticity 196 N . The strings are light and air resistance may be neglected.
7. Verify that \(P\) is in equilibrium when \(P\) is vertically below \(A\) and the length of the string \(P A\) is 1.5 m . The particle is set in motion along the line \(A B\) with both strings remaining taut. The displacement of \(P\) below the equilibrium position is denoted by \(x\) metres.
8. Show that the tension in the string \(P A\) is \(245 ( 0.7 + x )\) newtons, and the tension in the string \(P B\) is \(245 ( 0.3 - x )\) newtons.
9. Show that the motion of \(P\) is simple harmonic.
10. Given that the amplitude of the motion is 0.25 m , find the proportion of time for which \(P\) is above the mid-point of \(A B\).

3 An elastic rope has natural length 25 m and modulus of elasticity 980 N . One end of the rope is attached to a fixed point O , and a rock of mass 5 kg is attached to the other end; the rock is always vertically below O.

1. Find the extension of the rope when the rock is hanging in equilibrium. When the rock is moving with the rope stretched, its displacement is \(x\) metres below the equilibrium position at time \(t\) seconds.
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 7.84 x\). The rock is released from a position where the rope is slack, and when the rope just becomes taut the speed of the rock is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance below the equilibrium position at which the rock first comes instantaneously to rest.
4. Find the maximum speed of the rock.
5. Find the time between the rope becoming taut and the rock first coming to rest.
6. State three modelling assumptions you have made in answering this question.

3 A block of mass 200 kg is connected to a horizontal ceiling by four identical light elastic ropes, each having natural length 7 m and stiffness \(180 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). It is also connected to the floor by a single light elastic rope having stiffness \(80 \mathrm { Nm } ^ { - 1 }\). Throughout this question you may assume that all five ropes are stretched and vertical, and you may neglect air resistance. \begin{figure}[h] \end{figure} Fig. 3 shows the block resting in equilibrium, with each of the top ropes having length 10 m and the bottom rope having length 8 m .

1. Find the tension in one of the top ropes.
2. Find the natural length of the bottom rope. The block now moves vertically up and down. At time \(t\) seconds, the block is \(x\) metres below its equilibrium position.
3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x\). The motion is started by pulling the block down 2.2 m below its equilibrium position and releasing it from rest. The block then executes simple harmonic motion with amplitude 2.2 m .
4. Find the maximum magnitude of the acceleration of the block.
5. Find the speed of the block when it has travelled 3.8 m from its starting point.
6. Find the distance travelled by the block in the first 5 s .

3 A bungee jumper of mass 75 kg is connected to a fixed point A by a light elastic rope with stiffness \(300 \mathrm { Nm } ^ { - 1 }\). The jumper starts at rest at A and falls vertically. The lowest point reached by the jumper is 40 m vertically below A. Air resistance may be neglected.

1. Find the natural length of the rope.
2. Show that, when the rope is stretched and its extension is \(x\) metres, \(\ddot { x } + 4 x = 9.8\). The substitution \(y = x - c\), where \(c\) is a constant, transforms this equation to \(\ddot { y } = - 4 y\).
3. Find \(c\), and state the maximum value of \(y\).
4. Using standard simple harmonic motion formulae, or otherwise, find
   (A) the maximum speed of the jumper,
   (B) the maximum deceleration of the jumper.
5. Find the time taken for the jumper to fall from A to the lowest point.

2 A light elastic string AB has stiffness \(k\). The end A is attached to a fixed point and a particle of mass \(m\) is attached at the end B . With the string vertical, the particle is released from rest from a point at a distance \(a\) below its equilibrium position. At time \(t\), the displacement of the particle below the equilibrium position is \(x\) and the velocity of the particle is \(v\).

1. Show that $$m v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - k x$$
2. Show that, while the particle is moving upwards and the string is taut, $$v = - \sqrt { \frac { k } { m } \left( a ^ { 2 } - x ^ { 2 } \right) }$$
3. Hence use integration to find an expression for \(x\) at time \(t\) while the particle is moving upwards and the string is taut.

2 A light elastic string has natural length \(a\) and modulus of elasticity \(k m g\), where \(k > 2\). One end of the string is attached to a fixed point O . A particle P of mass \(m\) is attached to the other end of the string. P is held at rest a distance \(\frac { 3 } { 2 } a\) vertically below O . At time \(t\) after P is released, its vertical distance below O is \(y\).

1. Show that, while the string is in tension, the equation of motion of P is given by the differential equation \(\frac { d ^ { 2 } y } { d t ^ { 2 } } = ( k + 1 ) g - \frac { k g } { a } y\). A student transforms the differential equation in part (a) into the standard SHM equation \(\frac { d ^ { 2 } x } { d t ^ { 2 } } = - \omega ^ { 2 } x\).
2. - Find an expression for \(x\) in terms of \(y , k\) and \(a\).
   • Find an expression for \(\omega\) in terms of \(k , a\) and \(g\).

7 \begin{figure}[h] \end{figure} A particle P of mass \(m\) is attached to one end of a light elastic string of natural length \(6 a\) and modulus of elasticity 3 mg . The other end of the string is fixed to a point O on a smooth plane, which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The string lies along a line of greatest slope of the plane and P rests in equilibrium on the inclined plane at a point A , as shown in Fig. 7. P is now pulled a further distance \(2 a\) down the line of greatest slope through A and released from rest. At time \(t\) later, the displacement of P from A is \(x\), where the positive direction of \(x\) is down the plane.

1. Show that, until the string slackens, \(x\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + \frac { g x } { 2 a } = 0$$
2. Determine, in terms of \(a\) and \(g\), the time at which the string slackens.
3. Find, in terms of \(a\) and \(g\), the speed of P when the string slackens.

3. A light elastic string, of natural length \(l \mathrm {~m}\) and modulus of elasticity 14 N , is hanging vertically with its upper end fixed and a particle of mass \(m \mathrm {~kg}\) attached to the lower end. The particle is initially in equilibrium and air resistance is to be neglected.

1. Find, in terms of \(m , g\) and \(l\), the extension, \(e\), of the string when the particle is in equilibrium. The particle is pulled vertically downwards a further distance from its equilibrium position and released. In its subsequent motion, the string remains taut. Let \(x \mathrm {~m}\) denote the extension of the string from the equilibrium position at time \(t \mathrm {~s}\).
2. 1. Write down, in terms of \(x , m , g\) and \(l\), an expression for the tension in the string.
   2. Hence, show that the particle is moving with Simple Harmonic Motion which satisfies the differential equation, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 14 } { m l } x$$
   3. State the maximum distance that the particle could be pulled vertically downwards from its equilibrium position and still move with Simple Harmonic Motion. Give a reason for your answer.
3. Given that \(m = 0.5 , l = 0.7\) and that the particle is pulled to the position where \(x = 0.2\) before being released,
   1. find the maximum speed of the particle,
   2. determine the time taken for the particle to reach \(x = 0.15\) for the first time.

2. A particle \(P\) moves along the \(x\)-axis such that its position \(x\) metres, after \(t\) seconds, is given by $$x = \sin ( \pi t ) + \sqrt { 3 } \cos ( \pi t )$$

1. 1. Show that the motion of the particle \(P\) is Simple Harmonic. State the value of \(x\) at the centre of motion.
   2. Show that the period of the motion of \(P\) is 2 s and determine the amplitude. Suppose that another particle \(Q\) is introduced so that it also moves along the \(x\)-axis with Simple Harmonic Motion with centre of motion, \(O\), and period equal to that of particle \(P\). When \(t = 0\), the particle \(Q\) is at \(O\) and when it is \(2 \sqrt { 3 } \mathrm {~m}\) from \(O\) its speed is \(2 \pi \mathrm {~ms} ^ { - 1 }\).
2. Find the amplitude of particle \(Q\).
3. Determine the time when particles \(P\) and \(Q\) first meet.

6. The diagram shows a playground ride consisting of a seat \(P\), of mass 12 kg , attached to a vertical spring, which is fixed to a horizontal board. When the ride is at rest with nobody on it, the compression of the spring is 0.05 m .
\includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-4_305_654_1032_667} The spring is of natural length 0.75 m and modulus of elasticity \(\lambda\).

1. Find the value of \(\lambda\). The seat \(P\) is now pushed vertically downwards a further 0.05 m and is then released from rest.
2. Show that \(P\) makes Simple Harmonic oscillations of period \(\frac { \pi } { 7 }\) and write down the amplitude of the motion.
3. Find the maximum speed of \(P\).
4. Calculate the speed of \(P\) when it is at a distance 0.03 m from the equilibrium position.
5. Find the distance of \(P\) from the equilibrium position 1.6 s after it is released.[3]
6. State one modelling assumption you have made about the seat and one modelling assumption you have made about the spring.

1. The points \(A\) and \(B\) lie on a smooth horizontal surface with \(A B = 4.5 \mathrm {~m}\).

A light elastic string has natural length 1.5 m and modulus of elasticity 15 N . One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). A particle, \(P\), of mass 0.2 kg , is attached to the stretched string so that \(A P B\) is a straight line and \(A P = 1.5 \mathrm {~m}\). The particle rests in equilibrium on the surface. The particle is now moved directly towards \(A\) and is held on the surface so \(A P B\) is a straight line with \(A P = 1 \mathrm {~m}\). The particle is released from rest.

1. Prove that \(P\) moves with simple harmonic motion.
2. Find
   1. the maximum speed of \(P\) during the motion,
   2. the maximum acceleration of \(P\) during the motion.
3. Find the total time, in each complete oscillation of \(P\), for which the speed of \(P\) is greater than \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1 The displacement of a particle from its equilibrium position is \(x\) metres at time \(t\) seconds. The motion of the particle obeys the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 9 x$$ Calculate the period of its motion in seconds.
Circle your answer.
[0pt] [1 mark]
\(\frac { \pi } { 9 }\)
\(\frac { 2 \pi } { 9 }\)
\(\frac { \pi } { 3 }\)
\(\frac { 2 \pi } { 3 }\)