Simple Harmonic Motion

A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\) with period 2 s. At time \(t\) seconds the speed of \(P\) is \(v\) m s\(^{-1}\). When \(t = 0\), \(v = 0\) and \(P\) is at a point \(A\) where \(OA = 0.25\) m. Find the smallest positive value of \(t\) for which \(AP = 0.375\) m. [6]

1. A particle \(P\) moves in a straight line with simple harmonic motion between two fixed points \(A\) and \(B\). The particle performs 2 complete oscillations per second. The midpoint of \(A B\) is \(O\) and the midpoint of \(O A\) is \(C\)

The length of \(A B\) is 0.6 m .

1. Find the maximum speed of \(P\)
2. Find the time taken by \(P\) to move directly from \(O\) to \(C\)

\includegraphics{figure_5} A light elastic band, of total natural length \(a\) and modulus of elasticity \(\frac{1}{2}mg\), is stretched over two small smooth pins fixed at the same horizontal level and at a distance \(a\) apart. A particle of mass \(m\) is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle \(\beta\) with the vertical (see diagram). Express the tension in the band in terms of \(m\), \(g\) and \(\beta\), and hence show that \(\beta = \frac{1}{4}\pi\). [4] The particle is given a velocity of magnitude \(\sqrt{(ag)}\) vertically downwards. At time \(t\) the displacement of the particle from its equilibrium position is \(x\). Show that, neglecting air resistance, $$\ddot{x} = -\frac{2g}{a}x.$$ [3] Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time. [6]

1. Find the value of \(k\).
2. The particle \(P\) is released from rest at a point between \(A\) and \(B\) where both strings are taut. Show that \(P\) performs simple harmonic motion and state the period of the motion.
3. In the case where \(P\) is released from rest at a distance \(0.2 a \mathrm {~m}\) from \(M\), the speed of \(P\) is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(P\) is \(0.05 a \mathrm {~m}\) from \(M\). Find the value of \(a\).

7. \begin{figure}[h] \end{figure} Two points \(A\) and \(B\) lie on a smooth horizontal table where \(A B = 41\).
A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length I and modulus of elasticity 2 mg . The other end of the spring is attached to A . The particle P is also attached to one end of another light elastic spring of natural length I and modulus of elasticity mg . The other end of the spring is attached to B.
The particle \(P\) rests in equilibrium on the table at the point 0 , where \(A 0 = \frac { 5 } { 3 } I\), as shown in Figure 7.
The particle \(P\) is moved a distance \(\frac { 1 } { 2 } \mathrm { I }\) along the table, from 0 towards \(A\), and released from rest.

1. Show that P moves with simple harmonic motion of period T , where $$\mathrm { T } = 2 \pi \sqrt { \frac { l } { 3 g } }$$
2. Find, in terms of I and g , the speed of P as it passes through 0 .
3. Find, in terms of g , the maximum acceleration of P .
4. Find the exact time, in terms of I and g , from the instant when P is released from rest to the instant when P is first moving with speed \(\frac { 3 } { 4 } \sqrt { g l }\) \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-20_2269_56_311_1978} \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
   VILU SIHIL NI GLIUM ION OC
   VEYV SIHI NI ELIUM ION OC

7. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) are 4.2 m apart on a smooth horizontal floor. One end of a light elastic spring, of natural length 1.8 m and modulus of elasticity 20 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic spring, of natural length 0.9 m and modulus of elasticity 15 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 5.

1. Show that \(A O = 2.7 \mathrm {~m}\). The particle \(P\) now receives an impulse acting in the direction \(O B\) and moves away from \(O\) towards \(B\). In the subsequent motion \(P\) does not reach \(B\).
2. Show that \(P\) moves with simple harmonic motion about centre \(O\). The mass of \(P\) is 10 kg and the magnitude of the impulse is \(J \mathrm { Ns }\). Given that \(P\) first comes to instantaneous rest at the point \(C\) where \(A C = 2.9 \mathrm {~m}\),
3. 1. find the value of \(J\),
   2. find the time taken by \(P\) to travel a total distance of 0.5 m from when it first leaves \(O\).

1. Show that \(\theta = 0\) is a position of stable equilibrium.
2. Show that the kinetic energy of the system is \(4 m a ^ { 2 } \dot { \theta } ^ { 2 }\).
3. By differentiating the energy equation, then making suitable approximations for \(\sin \theta\) and \(\cos \theta\), find the approximate period of small oscillations about the equilibrium position \(\theta = 0\). \section*{[Question 7 is printed overleaf.]}

1. The diagram shows a uniform solid right circular cone of mass \(m \mathrm {~kg}\), height \(h \mathrm {~m}\) and base radius \(r \mathrm {~m}\) suspended by two vertical strings attached to the points \(P\) and \(Q\) on the circumference of the base. The vertex \(O\) of the cone is vertically below \(P\).
   1. Show that the tension in the string attached at \(Q\) is \(\frac { 3 m g } { 8 } \mathrm {~N}\). \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_296_277_269_1668}
   2. Find, in terms of \(m\) and \(g\), the tension in the other string.
   3. Two identical particles \(P\) and \(Q\) are connected by a light inextensible string passing through a small smooth-edged hole in a smooth table, as shown. \(P\) moves on the table in a horizontal circle of radius 0.2 m and \(Q\) hangs at rest. \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_309_430_859_1476}
   4. Calculate the number of revolutions made per minute by \(P\).
      (5 marks) \(Q\) is now also made to move in a horizontal circle of radius 0.2 m below the table. The part of the string between \(Q\) and the table makes an angle of \(45 ^ { \circ }\) with the vertical.
   5. Show that the numbers of revolutions per minute made by \(P\) and \(Q\) respectively are in the ratio \(2 ^ { 1 / 4 } : 1\). \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_293_428_1213_1499}
   6. A particle \(P\) of mass \(m \mathrm {~kg}\) is fixed to one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(k m g \mathrm {~N}\). The other end of the string is fixed to a point \(X\) on a horizontal plane. \(P\) rests at \(O\), where \(O X = l \mathrm {~m}\), with the string just taut. It is then pulled away from \(X\) through a distance \(\frac { 3 l } { 4 } \mathrm {~m}\) and released from rest. On this side of \(O\), the plane is smooth.
   7. Show that, as long as the string is taut, \(P\) performs simple harmonic motion.
   8. Given that \(P\) first returns to \(O\) with speed \(\sqrt { } ( g l ) \mathrm { ms } ^ { - 1 }\), find the value of \(k\).
   9. On the other side of \(O\) the plane is rough, the coefficient of friction between \(P\) and the plane being \(\mu\). If \(P\) does not reach \(X\) in the subsequent motion, show that \(\mu > \frac { 1 } { 2 }\). ( 4 marks)
   10. If, further, \(\mu = \frac { 3 } { 4 }\), show that the time which elapses after \(P\) is released and before it comes to rest is \(\frac { 1 } { 24 } ( 9 \pi + 32 ) \sqrt { \frac { l } { g } }\) s.
       (6 marks)

A rigid body consists of a thin uniform rod \(A B\), of mass \(4 m\) and length \(6 a\), joined at \(B\) to a point on the circumference of a uniform circular disc, with centre \(O\), mass \(8 m\) and radius \(2 a\). The point \(C\) on the circumference of the disc is such that \(B C\) is a diameter and \(A B C\) is a straight line (see diagram). The body rotates about a smooth fixed horizontal axis through \(C\), perpendicular to the plane of the disc. The angle between \(C A\) and the downward vertical at time \(t\) is denoted by \(\theta\).

1. Given that the body is performing small oscillations about the downward vertical, show that the period of these oscillations is approximately \(16 \pi \sqrt { } \left( \frac { a } { 11 g } \right)\).
2. Given instead that the body is released from rest in the position given by \(\cos \theta = 0.6\), find the maximum speed of \(A\).

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 \(t = 0.2\).

7. \begin{figure}[h] \end{figure} A smooth hollow narrow tube of length \(l\) has one open end and one closed end. The tube is fixed in a vertical position with the closed end at the bottom. A light elastic spring has natural length \(l\) and modulus of elasticity \(8 m g\). The spring is inside the tube and has one end attached to a fixed point \(O\) on the closed end of the tube. The other end of the spring is attached to a particle \(P\) of mass \(m\). The particle rests in equilibrium at a distance \(e\) below the top of the tube, as shown in Figure 4.

1. Find \(e\) in terms of \(l\). The particle \(P\) is now held inside the tube at a distance \(\frac { 1 } { 2 } l\) below the top of the tube and released from rest at time \(t = 0\)
2. Prove that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \left( \frac { l } { 8 g } \right) }\). The particle \(P\) passes through the open top of the tube with speed \(u\).
3. Find \(u\) in terms of \(g\) and \(l\).
4. Find the time taken for \(P\) to first attain a speed of \(\sqrt { \left( \frac { 9 g l } { 32 } \right) }\).

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length 2l. The other end of the spring is attached to a fixed point \(A\). The particle \(P\) hangs in equilibrium vertically below \(A\), at the point \(E\) where \(A E = 6 l\). The particle \(P\) is then raised a vertical distance \(2 l\) and released from rest.

Air resistance is modelled as being negligible.

1. Show that \(P\) moves with simple harmonic motion of period \(T\) where $$T = 4 \pi \sqrt { \frac { l } { g } }$$
2. Find, in terms of \(m , l\) and \(g\), the kinetic energy of \(P\) as it passes through \(E\)
3. Find, in terms of \(T\), the exact time from the instant when \(P\) is released to the instant when \(P\) has moved a distance 31 .

1 A particle is moving with simple harmonic motion in a straight line. The period is 0.2 s and the amplitude of the motion is 0.3 m . Find the maximum speed and the maximum acceleration of the particle.

The movement of a particle is described by the simple harmonic equation $$\ddot{x} = -25x$$ where \(x\) metres is the displacement of the particle at time \(t\) seconds, and \(\ddot{x}\) m s\(^{-2}\) is the acceleration of the particle. The maximum displacement of the particle is 9 metres. Find the maximum speed of the particle. Circle your answer. [1 mark] \(15\) m s\(^{-1}\) \quad\quad \(45\) m s\(^{-1}\) \quad\quad \(75\) m s\(^{-1}\) \quad\quad \(135\) m s\(^{-1}\)

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 elastic string of modulus of elasticity \(8 m g\) and natural length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is pulled vertically downwards a distance \(\frac { 1 } { 4 } a\) from its equilibrium position and released from rest. Show that the string first becomes slack after a time \(\frac { 2 \pi } { 3 } \sqrt { } \left( \frac { a } { 8 g } \right)\). Find, in terms of \(a\), the total distance travelled by \(P\) from its release until it subsequently comes to instantaneous rest for the first time.

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]

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\). The other end of the string is attached to a fixed point on a ceiling. The particle \(P\) hangs in equilibrium at a distance \(D\) below the ceiling.

The particle \(P\) is now pulled vertically downwards until it is a distance \(3 l\) below the ceiling and released from rest. Given that \(P\) comes to instantaneous rest just before it reaches the ceiling,

1. show that \(D = \frac { 5 l } { 3 }\)
2. Show that, while the elastic string is stretched, \(P\) moves with simple harmonic motion, with period \(2 \pi \sqrt { \frac { 2 l } { 3 g } }\)
3. Find, in terms of \(g\) and \(l\), the exact time from the instant when \(P\) is released to the instant when the elastic string first goes slack.

3 A particle \(P\) is performing simple harmonic motion with amplitude 0.25 m . During each complete oscillation, \(P\) moves with a speed that is less than or equal to half of its maximum speed for \(\frac { 4 } { 3 }\) seconds. Find the period of the motion and the maximum speed of \(P\).

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 $$\frac{d^2x}{dt^2} = -9x$$ Calculate the period of its motion in seconds. Circle your answer. [1 mark] \(\frac{\pi}{9}\) \(\quad\) \(\frac{2\pi}{9}\) \(\quad\) \(\frac{\pi}{3}\) \(\quad\) \(\frac{2\pi}{3}\)

An elastic string spring of modulus \(2mg\) and natural length \(l\) is fixed at one end. To the other end is attached a mass \(m\) which is allowed to hang in equilibrium. The mass is then pulled vertically downwards through a distance \(l\) and released from rest. The air resistance is modelled as having magnitude \(2m\omega v\), where \(v\) is the speed of the particle and \(\omega = \sqrt{\frac{g}{l}}\). The particle is at distance \(x\) from its equilibrium position at time \(t\).

1. Show that \(\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} + 2\omega \frac{\mathrm{d} x}{\mathrm{d} t} + 2\omega^2 x = 0\). [7]
2. Find the general solution of this differential equation. [4]
3. Hence find the period of the damped harmonic motion. [1]

2 A particle of mass \(m\) is attached to the mid-point of a light elastic string. The string is stretched between two points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 2 a\). The string has modulus of elasticity \(\lambda\) and natural length \(2 l\), where \(l < a\). The particle is in motion on the surface along a line passing through the mid-point of \(A B\) and perpendicular to \(A B\). When the displacement of the particle from \(A B\) is \(x\), the tension in the string is \(T\). Given that \(x\) is small enough for \(x ^ { 2 }\) to be neglected, show that $$T = \frac { \lambda } { l } ( a - l )$$ The particle is slightly disturbed from its equilibrium position. Show that it will perform approximate simple harmonic motion and find the period of the motion.

2 A particle of mass \(m\) is attached to the mid-point of a light elastic string. The string is stretched between two points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 2 a\). The string has modulus of elasticity \(\lambda\) and natural length \(2 l\), where \(l < a\). The particle is in motion on the surface along a line passing through the mid-point of \(A B\) and perpendicular to \(A B\). When the displacement of the particle from \(A B\) is \(x\), the tension in the string is \(T\). Given that \(x\) is small enough for \(x ^ { 2 }\) to be neglected, show that $$T = \frac { \lambda } { l } ( a - l )$$ The particle is slightly disturbed from its equilibrium position. Show that it will perform approximate simple harmonic motion and find the period of the motion.

7 \includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-4_557_1036_278_553} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is pivoted to a fixed point at \(A\) and is free to rotate in a vertical plane. Two fixed vertical wires in this plane are a distance \(6 a\) apart and the point \(A\) is half-way between the two wires. Light smooth rings \(R _ { 1 }\) and \(R _ { 2 }\) slide on the wires and are connected to \(B\) by light elastic strings, each of natural length \(a\) and modulus of elasticity \(\frac { 1 } { 4 } m g\). The strings \(B R _ { 1 }\) and \(B R _ { 2 }\) are always horizontal and the angle between \(A B\) and the upward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram).

1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a \left( 1 + \cos \theta + \sin ^ { 2 } \theta \right) .$$
2. Given that \(\theta = 0\) is a position of stable equilibrium, find the approximate period of small oscillations about this position.

1. Find an expression for the stiffness of the spring, \(k \mathrm { Nm } ^ { - 1 }\), in terms of \(m , h\) and \(g\). The particle is pushed down a further distance from the equilibrium position and released from rest. At time \(t\) seconds, the displacement of the particle from the equilibrium position of the system is \(y \mathrm {~m}\) in the downward direction, as shown in Fig. 5.3. You are given that \(| y | \leq h\).
2. Show that the motion of the particle is modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { g y } { h } = 0\).
3. Find an expression for the period of the motion of the particle.
4. Would the model for the motion of the particle be valid for large values of \(m\) ? Justify your answer.

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 \(a\).

A bungee jumper of mass \(m\) kg is attached to an elastic rope. The other end of the rope is attached to a fixed point. The bungee jumper falls vertically from the fixed point. At time \(t\) seconds after the rope first becomes taut, the extension of the rope is \(x\) metres and the speed of the bungee jumper is \(v\) m s\(^{-1}\)

1. A model for the motion while the rope remains taut assumes that the forces acting on the bungee jumper are • the weight of the bungee jumper • a tension in the rope of magnitude \(kx\) newtons • an air resistance force of magnitude \(Rv\) newtons where \(k\) and \(R\) are constants such that \(4km > R^2\)
   1. Show that this model gives the result $$x = e^{-\frac{Rt}{2m}} \left( A \cos \frac{\sqrt{4km - R^2}}{2m} t + B \sin \frac{\sqrt{4km - R^2}}{2m} t \right) + \frac{mg}{k}$$ where \(A\) and \(B\) are constants, and \(g\) m s\(^{-2}\) is the acceleration due to gravity. You do not need to find the value of \(A\) or the value of \(B\) [6 marks]
   2. It is also given that: \(k = 16\) \(R = 20\) \(m = 62.5\) \(g = 9.8\) m s\(^{-2}\) and that the speed of the bungee jumper when the rope becomes taut is 14 m s\(^{-1}\) Show that, to the nearest integer, \(A = -38\) and \(B = 16\) [6 marks]
2. A second, simpler model assumes that the air resistance is zero. The values of \(k\), \(m\) and \(g\) remain the same. Find an expression for \(x\) in terms of \(t\) according to this simpler model, giving the values of all constants to two significant figures. [4 marks]

\includegraphics{figure_3} A small object \(P\) is attached to one end of each of two vertical light elastic strings. One string is of natural length \(0.4\) m and has modulus of elasticity \(10\) N; the other string is of natural length \(0.5\) m and has modulus of elasticity \(12\) N. The upper ends of both strings are attached to a fixed horizontal beam and \(P\) hangs in equilibrium \(0.6\) m below the beam (see diagram).

1. Show that the weight of \(P\) is \(7.4\) N and find the total elastic potential energy stored in the two strings when \(P\) is hanging in equilibrium. [6]

\(P\) is then held at a point \(0.7\) m below the beam with the strings vertical. \(P\) is released from rest.

1. Show that, throughout the subsequent motion, \(P\) performs simple harmonic motion, and find the period. [7]

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 .

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.

A particle \(P\) of mass \(0.2\) kg oscillates with simple harmonic motion between the points \(A\) and \(B\), coming to rest at both points. The distance \(AB\) is \(0.2\) m, and \(P\) completes \(5\) oscillations every second.

1. Find, to \(3\) significant figures, the maximum resultant force exerted on \(P\). [6]

When the particle is at \(A\), it is struck a blow in the direction \(BA\). The particle now oscillates with simple harmonic motion with the same frequency as previously but twice the amplitude.

1. Find, to \(3\) significant figures, the speed of the particle immediately after it has been struck. [5]

A particle \(P\) of mass \(0.2\) kg oscillates with simple harmonic motion between the points \(A\) and \(B\), coming to rest at both points. The distance \(AB\) is \(0.2\) m, and \(P\) completes \(5\) oscillations every second.

1. Find, to \(3\) significant figures, the maximum resultant force exerted on \(P\). [6]

When the particle is at \(A\), it is struck a blow in the direction \(BA\). The particle now oscillates with simple harmonic motion with the same frequency as previously but twice the amplitude.

1. Find, to \(3\) significant figures, the speed of the particle immediately after it has been struck. [5]

7 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-4_588_629_274_758} A particle \(P\) of mass 0.8 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.4 m . A particle \(Q\) is suspended from \(O\) by an identical string. With the string \(O P\) taut and inclined at \(\frac { 1 } { 3 } \pi\) radians to the vertical, \(P\) is projected with speed \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the string so as to strike \(Q\) directly (see diagram). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 7 }\).

1. Calculate the tension in the string immediately after \(P\) is set in motion.
2. Immediately after \(P\) and \(Q\) collide they have equal speeds and are moving in opposite directions. Show that \(Q\) starts to move with speed \(0.15 \mathrm {~ms} ^ { - 1 }\).
3. Prove that before the second collision between \(P\) and \(Q , Q\) is moving with approximate simple harmonic motion.
4. Hence find the time interval between the first and second collisions of \(P\) and \(Q\).

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.

5 A particle of mass \(m\) is attached by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda\) to a fixed point \(A\), from which it is allowed to fall freely. The particle first comes to rest, instantaneously, at \(B\), where \(A B = 2 l\). Prove that

1. \(\lambda = 4 m g\),
2. while the string is taut, \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g } { l } x\), where \(x\) is the displacement from the equilibrium position at time \(t\),
3. the time taken between the first occasion when the string becomes taut and the next occasion when it becomes slack is $$\left[ \frac { 1 } { 2 } \pi + \sin ^ { - 1 } \left( \frac { 1 } { 3 } \right) \right] \sqrt { \frac { l } { g } }$$

The point \(O\) is on the fixed horizontal line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(OA = 0.1\) m and \(OB = 0.5\) m, with \(B\) between \(O\) and \(A\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The kinetic energy of \(P\) when it is at \(A\) is twice its kinetic energy when it is at \(B\). Find the amplitude of the motion. [3]

1. In practice the minimum distance predicted by the model is incorrect. Is the minimum distance predicted by the model likely to be too big or too small? Explain your answer with reference to the model. [2 marks]

1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds the displacement of \(P\) from the origin \(O\) is \(x\) metres, where \(x = 4 \cos \left( \frac { 1 } { 5 } \pi t \right)\)
   1. Prove that \(P\) is moving with simple harmonic motion.
   2. Find the period of the motion.
   3. State the amplitude of the motion.
   4. Find, in terms of \(\pi\), the maximum speed of \(P\)
   The points \(A\) and \(B\) lie on the \(x\)-axis, on opposite sides of \(O\), with \(O A = 1.5 \mathrm {~m}\) and \(O B = 2.5 \mathrm {~m}\).
2. Find the time taken by \(P\) to move directly from \(A\) to \(B\).

4 A particle is connected to a fixed point by a light inextensible string of length 2.45 m to make a simple pendulum. The particle is released from rest with the string taut and inclined at 0.1 radians to the downward vertical.

1. Show that the motion of the particle is approximately simple harmonic with period 3.14 s , correct to 3 significant figures. Calculate, in either order,
2. the angular speed of the pendulum when it has moved 0.04 radians from the initial position,
3. the time taken by the pendulum to move 0.04 radians from the initial position.

A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs freely at rest in equilibrium at the point \(E\). The distance of \(E\) below \(A\) is \((l + e)\).

1. Find \(e\) in terms of \(l\). [2]

At time \(t = 0\), the particle is projected vertically downwards from \(E\) with speed \(\sqrt{gl}\).

1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion. [5]
2. Find the amplitude of the simple harmonic motion. [3]
3. Find the time at which the string first goes slack. [4]

A particle \(P\) oscillates in simple harmonic motion between the points \(A\) and \(B\), where \(AB = 6\) m. The period of the motion is \(4\pi\) s. Find the speed of \(P\) when it is 2 m from \(B\). [3]

2. A particle \(P\) moves with simple harmonic motion and comes to rest at two points \(A\) and \(B\) which are 0.24 m apart on a horizontal line. The time for \(P\) to travel from \(A\) to \(B\) is 1.5 s . The midpoint of \(A B\) is \(O\). At time \(t = 0 , P\) is moving through \(O\), towards \(A\), with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(u\).
2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
3. Find the speed of \(P\) when \(t = 2 \mathrm {~s}\).

4 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-2_364_1313_1224_376} An unidentified aircraft \(U\) is flying horizontally with constant velocity \(250 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(040 ^ { \circ }\). Two spotter planes \(P\) and \(Q\) are flying horizontally at the same height as \(U\), and at one instant \(P\) is 15000 m due west of \(U\), and \(Q\) is 15000 m due east of \(U\) (see diagram).

1. Plane \(P\) is flying with constant velocity \(210 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(070 ^ { \circ }\).

1. A particle \(P\) is moving in a straight line with simple harmonic motion of period 4 s . The centre of the motion is the point \(O\)

At time \(t = 0 , P\) passes through \(O\) At time \(t = 0.5 \mathrm {~s} , P\) is moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Show that the amplitude of the motion is \(\frac { 4 \sqrt { 2 } } { \pi } \mathrm {~m}\)
2. Find the maximum speed of \(P\)

5. A piston in a machine is modelled as a particle of mass 0.2 kg attached to one end \(A\) of a light elastic spring, of natural length 0.6 m and modulus of elasticity 48 N . The other end \(B\) of the spring is fixed and the piston is free to move in a horizontal tube which is assumed to be smooth. The piston is released from rest when \(A B = 0.9 \mathrm {~m}\).

1. Prove that the motion of the piston is simple harmonic with period \(\frac { \pi } { 10 } \mathrm {~s}\).
   (5)
2. Find the maximum speed of the piston.
   (2)
3. Find, in terms of \(\pi\), the length of time during each oscillation for which the length of the spring is less than 0.75 m .
   (5)

A particle \(P\), of mass \(0.2\) kg, moves in simple harmonic motion along a straight line under the action of a resultant force of magnitude \(F\) N. The distance between the end-points of the motion is \(0.6\) m, and the period of the motion is \(0.5\) s. Find the greatest value of \(F\) during the motion. [5]

A light spring has natural length \(a\) and modulus of elasticity \(k m g\). The spring lies on a smooth horizontal surface with one end attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The system is in equilibrium with \(O P = a\). The particle is projected towards \(O\) with speed \(u\) and comes to instantaneous rest when \(O P = \frac { 3 } { 4 } a\).

1. Use an energy method to show that \(k = \frac { 16 u ^ { 2 } } { a g }\).
2. Show that \(P\) performs simple harmonic motion and find the period of this motion, giving your answer in terms of \(u\) and \(a\).
3. Find, in terms of \(u\) and \(a\), the time that elapses before \(P\) first loses \(25 \%\) of its initial kinetic energy.