4.10f Simple harmonic motion: x'' = -omega^2 x

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.

1

1. 1. Write down the dimensions of force. The period, \(t\), of a vibrating wire depends on its tension, \(F\), its length, \(l\), and its mass per unit length, \(\sigma\).
   2. Assuming that the relationship is of the form \(t = k F ^ { \alpha } l ^ { \beta } \sigma ^ { \gamma }\), where \(k\) is a dimensionless constant, use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\). Two lengths are cut from a reel of uniform wire. The first has length 1.2 m , and it vibrates under a tension of 90 N . The second has length 2.0 m , and it vibrates with the same period as the first wire.
   3. Find the tension in the second wire. (You may assume that changing the tension does not significantly change the mass per unit length.)
2. The midpoint M of a vibrating wire is moving in simple harmonic motion in a straight line, with amplitude 0.018 m and period 0.01 s .
   1. Find the maximum speed of M .
   2. Find the distance of M from the centre of the motion when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 A light elastic rope has natural length 15 m . One end of the rope is attached to a fixed point O and the other end is attached to a small rock of mass 12 kg . When the rock is hanging in equilibrium vertically below O , the length of the rope is 15.8 m .

1. Show that the modulus of elasticity of the rope is 2205 N . The rock is pulled down to the point 20 m vertically below O , and is released from rest in this position. It moves upwards, and comes to rest instantaneously, with the rope slack, at the point A .
2. Find the acceleration of the rock immediately after it is released.
3. Use an energy method to find the distance OA. At time \(t\) seconds after release, the rope is still taut and the displacement of the rock below the equilibrium position is \(x\) metres.
4. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 12.25 x\).
5. Write down an expression for \(x\) in terms of \(t\), and hence find the time between releasing the rock and the rope becoming slack.

3 Ben has mass 60 kg and he is considering doing a bungee jump using an elastic rope with natural length 32 m . One end of the rope is attached to a fixed point O , and the other end is attached to Ben. When Ben is supported in equilibrium by the rope, the length of the rope is 32.8 m . To predict what will happen, Ben is modelled as a particle B, the rope is assumed to be light, and air resistance is neglected. B is released from rest at O and falls vertically. When the rope becomes stretched, \(x \mathrm {~m}\) denotes the extension of the rope.

1. Find the stiffness of the rope.
2. Use an energy argument to show that, when B comes to rest instantaneously with the rope stretched, $$x ^ { 2 } - 1.6 x - 51.2 = 0$$ Hence find the length of the rope when B is at its lowest point.
3. Show that, while the rope is stretched, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12.25 x = 9.8$$ where \(t\) is the time measured in seconds.
4. Find the time taken for B to travel between the equilibrium position \(( x = 0.8 )\) and the lowest point.
5. Find the acceleration of \(\mathbf { B }\) when it is at the lowest point, and comment on the implications for Ben.

3 A particle is oscillating in a vertical line. At time \(t\) seconds, its displacement above the centre of the oscillations is \(x\) metres, where \(x = A \sin \omega t + B \cos \omega t\) (and \(A , B\) and \(\omega\) are constants).

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \omega ^ { 2 } x\). When \(t = 0\), the particle is 2 m above the centre of the oscillations, the velocity is \(1.44 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards, and the acceleration is \(0.18 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards.
2. Find \(A , B\) and \(\omega\).
3. Show that the period of oscillation is 20.9 s (correct to 3 significant figures), and find the amplitude.
4. Find the total distance travelled by the particle between \(t = 12\) and \(t = 24\).

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.

4 Fig. 4 shows a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Two fixed points A and B on the plane are 4.55 m apart with B higher than A on a line of greatest slope. A particle P of mass 0.25 kg is in contact with the plane and is connected to A and to B by two light elastic strings. The string AP has natural length 1.5 m and modulus of elasticity 7.35 N ; the string BP has natural length 2.5 m and modulus of elasticity 7.35 N . The particle P moves along part of the line AB , with both strings taut throughout the motion. \begin{figure}[h] \end{figure}

1. Show that, when \(\mathrm { AP } = 1.55 \mathrm {~m}\), the acceleration of P is zero.
2. Taking \(\mathrm { AP } = ( 1.55 + x ) \mathrm { m }\), write down the tension in the string AP , in terms of \(x\), and show that the tension in the string BP is \(( 1.47 - 2.94 x ) \mathrm { N }\).
3. Show that the motion of P is simple harmonic, and find its period. The particle P is released from rest with \(\mathrm { AP } = 1.5 \mathrm {~m}\).
4. Find the time after release when P is first moving down the plane with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

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.

1

1. A particle P is executing simple harmonic motion, and the centre of the oscillations is at the point O . The maximum speed of P during the motion is \(5.1 \mathrm {~ms} ^ { - 1 }\). When P is 6 m from O , its speed is \(4.5 \mathrm {~ms} ^ { - 1 }\). Find the period and the amplitude of the motion.
2. The force \(F\) of gravitational attraction between two objects of masses \(m _ { 1 }\) and \(m _ { 2 }\) at a distance \(d\) apart is given by \(F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }\), where \(G\) is the universal gravitational constant.
   1. Find the dimensions of \(G\). Three objects, each of mass \(m\), are moving in deep space under mutual gravitational attraction. They move round a single circle with constant angular speed \(\omega\), and are always at the three vertices of an equilateral triangle of side \(R\). You are given that \(\omega = k G ^ { \alpha } m ^ { \beta } R ^ { \gamma }\), where \(k\) is a dimensionless constant.
   2. Find \(\alpha , \beta\) and \(\gamma\). For three objects of mass 2500 kg at the vertices of an equilateral triangle of side 50 m , the angular speed is \(2.0 \times 10 ^ { - 6 } \mathrm { rad } \mathrm { s } ^ { - 1 }\).
   3. Find the angular speed for three objects of mass \(4.86 \times 10 ^ { 14 } \mathrm {~kg}\) at the vertices of an equilateral triangle of side 30000 m .

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 .

1

1. 1. Write down the dimensions of the following quantities. \begin{displayquote} Velocity
      Acceleration
      Force
      Density (which is mass per unit volume)
      Pressure (which is force per unit area) \end{displayquote} For a fluid with constant density \(\rho\), the velocity \(v\), pressure \(P\) and height \(h\) at points on a streamline are related by Bernoulli's equation $$P + \frac { 1 } { 2 } \rho v ^ { 2 } + \rho g h = \mathrm { constant } ,$$ where \(g\) is the acceleration due to gravity.
   2. Show that the left-hand side of Bernoulli's equation is dimensionally consistent.
2. In a wave tank, a float is performing simple harmonic motion with period 3.49 s in a vertical line. The height of the float above the bottom of the tank is \(h \mathrm {~m}\) at a time \(t \mathrm {~s}\). When \(t = 0\), the height has its maximum value. The value of \(h\) varies between 1.6 and 2.2.
   1. Sketch a graph showing how \(h\) varies with \(t\).
   2. Express \(h\) in terms of \(t\).
   3. Find the magnitude and direction of the acceleration of the float when \(h = 1.7\).

3 A small block B has mass 2.5 kg . A light elastic string connects B to a fixed point P , and a second light elastic string connects \(B\) to a fixed point \(Q\), which is 6.5 m vertically below \(P\). The string PB has natural length 3.2 m and stiffness \(35 \mathrm { Nm } ^ { - 1 }\); the string BQ has natural length 1.8 m and stiffness \(5 \mathrm { Nm } ^ { - 1 }\). The block B is released from rest in the position 4.4 m vertically below P . You are given that B performs simple harmonic motion along part of the line PQ, and that both strings remain taut throughout the motion. Air resistance may be neglected. At time \(t\) seconds after release, the length of the string PB is \(x\) metres (see Fig. 3). \begin{figure}[h] \end{figure}

1. Find, in terms of \(x\), the tension in the string PB and the tension in the string BQ .
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 64 - 16 x\).
3. Find the value of \(x\) when B is at the centre of oscillation.
4. Find the period of oscillation.
5. Write down the amplitude of the motion and find the maximum speed of B.
6. Find the time after release when \(B\) is first moving downwards with speed \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3

1. 1. Write down the dimensions of velocity, force and density (which is mass per unit volume). A vehicle moving with velocity \(v\) experiences a force \(F\), due to air resistance, given by $$F = \frac { 1 } { 2 } C \rho ^ { \alpha } v ^ { \beta } A ^ { \gamma }$$ where \(\rho\) is the density of the air, \(A\) is the cross-sectional area of the vehicle, and \(C\) is a dimensionless quantity called the drag coefficient.
   2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. A light rod is freely pivoted about a fixed point at one end and has a heavy weight attached to its other end. The rod with the weight attached is oscillating in a vertical plane as a simple pendulum with period 4.3 s . The maximum angle which the rod makes with the vertical is 0.08 radians. You may assume that the motion is simple harmonic.
   1. Find the angular speed of the rod when it makes an angle of 0.05 radians with the vertical.
   2. Find the time taken for the pendulum to swing directly from a position where the rod makes an angle of 0.05 radians on one side of the vertical to the position where the rod makes an angle of 0.05 radians on the other side of the vertical.

4 A particle P is performing simple harmonic motion in a vertical line. At time \(t \mathrm {~s}\), its displacement \(x \mathrm {~m}\) above a fixed point O is given by $$x = A \sin \omega t + B \cos \omega t$$ where \(A , B\) and \(\omega\) are constants.

1. Show that the acceleration of P , in \(\mathrm { ms } ^ { - 2 }\), is \(- \omega ^ { 2 } x\). When \(t = 0 , \mathrm { P }\) is 16 m below O , moving with velocity \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards, and has acceleration \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards.
2. Find the values of \(A , B\) and \(\omega\).
3. Find the maximum displacement, the maximum speed, and the maximum acceleration of P .
4. Find the speed and the direction of motion of P when \(t = 15\).
5. Find the distance travelled by P between \(t = 0\) and \(t = 15\).

3 A particle Q is performing simple harmonic motion in a vertical line. Its height, \(x\) metres, above a fixed level at time \(t\) seconds is given by $$x = c + A \cos ( \omega t - \phi )$$ where \(c , A , \omega\) and \(\phi\) are constants.

1. Show that \(\ddot { x } = - \omega ^ { 2 } ( x - c )\). Fig. 3 shows the displacement-time graph of Q for \(0 \leqslant t \leqslant 14\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-4_547_1079_703_495} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Find exact values for \(c , A , \omega\) and \(\phi\).
3. Find the maximum speed of Q .
4. Find the height and the velocity of Q when \(t = 0\).
5. Find the distance travelled by Q between \(t = 0\) and \(t = 14\).

3 A light spring, with modulus of elasticity 686 N , has one end attached to a fixed point A . The other end is attached to a particle P of mass 18 kg which hangs in equilibrium when it is 2.2 m vertically below A .

1. Find the natural length of the spring AP . Another light spring has natural length 2.5 m and modulus of elasticity 145 N . One end of this spring is now attached to the particle P , and the other end is attached to a fixed point B which is 2.5 m vertically below P (so leaving the equilibrium position of P unchanged). While in its equilibrium position, P is set in motion with initial velocity \(3.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, as shown in Fig. 3. It now executes simple harmonic motion along part of the vertical line AB . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-4_721_383_726_831} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} At time \(t\) seconds after it is set in motion, P is \(x\) metres below its equilibrium position.
2. Show that the tension in the spring AP is \(( 176.4 + 392 x ) \mathrm { N }\), and write down an expression for the thrust in the spring BP.
3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 25 x\).
4. Find the period and the amplitude of the motion.
5. Find the magnitude and direction of the velocity of P when \(t = 2.4\).
6. Find the total distance travelled by P during the first 2.4 seconds of its motion.

1

1. The speed \(v\) of sound in a solid material is given by \(v = \sqrt { \frac { E } { \rho } }\), where \(E\) is Young's modulus for the material and \(\rho\) is its density.
   1. Find the dimensions of Young's modulus. The density of steel is \(7800 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) and the speed of sound in steel is \(6100 \mathrm {~ms} ^ { - 1 }\).
   2. Find Young's modulus for steel, stating the units in which your answer is measured. A tuning fork has cylindrical prongs of radius \(r\) and length \(l\). The frequency \(f\) at which the tuning fork vibrates is given by \(f = k c ^ { \alpha } E ^ { \beta } \rho ^ { \gamma }\), where \(c = \frac { l ^ { 2 } } { r }\) and \(k\) is a dimensionless constant.
   3. Find \(\alpha , \beta\) and \(\gamma\).
2. A particle P is performing simple harmonic motion along a straight line, and the centre of the oscillations is O . The points X and Y on the line are on the same side of O , at distances 3.9 m and 6.0 m from O respectively. The speed of P is \(1.04 \mathrm {~ms} ^ { - 1 }\) when it passes through X and \(0.5 \mathrm {~ms} ^ { - 1 }\) when it passes through Y.
   1. Find the amplitude and the period of the oscillations.
   2. Find the time taken for P to travel directly from X to Y .

3 Fig. 3 shows the fixed points A and F which are 9.5 m apart on a smooth horizontal surface and points B and D on the line AF such that \(\mathrm { AB } = \mathrm { DF } = 3.0 \mathrm {~m}\). A small block of mass 10.5 kg is joined to A by a light elastic string of natural length 3.0 m and stiffness \(12 \mathrm { Nm } ^ { - 1 }\); the block is joined to F by a light elastic string of natural length 3.0 m and stiffness \(30 \mathrm { Nm } ^ { - 1 }\). The block is released from rest at B and then slides along part of the line AF . The block has zero acceleration when it is at a point C , and it comes to instantaneous rest at a point E . \begin{figure}[h] \end{figure}

1. Find the distance BC . At time \(t \mathrm {~s}\) the displacement of the block from C is \(x \mathrm {~m}\), measured in the direction AF .
2. Show that, when the block is between B and \(\mathrm { D } , \frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x\).
3. Find the maximum speed of the block.
4. Find the distance of the block from C when its speed is \(4.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
5. Find the time taken for the block to travel from B to D.
6. Find the distance DE .

3. A particle is performing simple harmonic motion along a straight line between the points \(A\) and \(B\) where \(A B = 8 \mathrm {~m}\). The period of the motion is 12 seconds.

1. Find the maximum speed of the particle in terms of \(\pi\). The points \(P\) and \(Q\) are on the line \(A B\) at distances of 3 m and 6 m respectively from \(A\).
2. Find, correct to 3 significant figures, the time it takes for the particle to travel directly from \(P\) to \(Q\).
   (6 marks)

4. On a particular day, high tide at the entrance to a harbour occurs at 11 a.m. and the water depth is 14 m . Low tide occurs \(6 \frac { 1 } { 4 }\) hours later at which time the water depth is 6 m . In a model of the situation, the water level is assumed to perform simple harmonic motion.
Using this model,

1. write down the amplitude and period of the motion. A ship needs a depth of 9 m before it can enter or leave the harbour.
2. Show that on this day a ship must enter the harbour by 2.38 p.m., correct to the nearest minute, or wait for low tide to pass.
   (6 marks)
   Given that a ship is not ready to enter the harbour until 5 p.m.,
3. find, to the nearest minute, how long the ship must wait before it can enter the harbour.

6. \begin{figure}[h] \end{figure} Figure 2 shows a particle \(P\) of mass \(m\) which lies on a smooth horizontal table. It is attached to a point \(A\) on the table by a light elastic spring of natural length \(3 a\) and modulus of elasticity \(\lambda\), and to a point \(B\) on the table by a light elastic spring of natural length \(2 a\) and modulus of elasticity \(2 \lambda\). The distance between the points \(A\) and \(B\) is \(7 a\).

1. Show that in equilibrium \(A P = \frac { 9 } { 2 } a\). The particle is released from rest at a point \(Q\) where \(Q\) lies on the line \(A B\) and \(A Q = 5 a\).
2. Prove that the subsequent motion of the particle is simple harmonic with a period of \(\pi \sqrt { \frac { 3 m a } { \lambda } }\).
   (9 marks)

4. A particle of mass 0.5 kg is moving on a straight line with simple harmonic motion. At time \(t = 0\) the particle is instantaneously at rest at the point \(A\). It next comes instantaneously to rest 3 seconds later at the point \(B\) where \(A B = 4 \mathrm {~m}\).

1. For the motion of the particle write down
   1. the period,
   2. the amplitude.
2. Find the maximum kinetic energy of the particle in terms of \(\pi\). The point \(C\) lies on \(A B\) at a distance of 1.2 m from \(B\).
3. Find the time it takes the particle to travel directly from \(A\) to \(C\), giving your answer in seconds correct to 2 decimal places.
   (4 marks)

4. A particle moves with simple harmonic motion along a straight line. When the particle is 3 cm from its centre of motion it has a speed of \(8 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) and an acceleration of magnitude \(12 \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).

1. Show that the period of the motion is \(\pi\) seconds.
2. Find the amplitude of the motion.
3. Hence, find the greatest speed of the particle.

6. A particle \(P\) of mass 2.5 kg is moving with simple harmonic motion in a straight line between two points \(A\) and \(B\) on a smooth horizontal table. When \(P\) is 3 m from \(O\), the centre of the oscillations, its speed is \(6 \mathrm {~ms} ^ { - 1 }\). When \(P\) is 2.25 m from \(O\), its speed is \(8 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(A B = 7.5 \mathrm {~m}\).
2. Find the period of the motion.
3. Find the kinetic energy of \(P\) when it is 2.7 m from \(A\).
4. Show that the time taken by \(P\) to travel directly from \(A\) to the midpoint of \(O B\) is \(\frac { \pi } { 4 }\) seconds.