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

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]

\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]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(3m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(O\) on a smooth plane that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac{3}{5}\). The system rests in equilibrium with \(P\) on the plane at the point \(E\). The length of the spring in this position is \(\frac{5}{4}a\).

1. Find the value of \(k\). [3]

The particle \(P\) is now replaced by a particle \(Q\) of mass \(2m\) and \(Q\) is released from rest at the point \(E\).

1. Show that, in the resulting motion, \(Q\) performs simple harmonic motion. State the centre and the period of the motion. [6]
2. Find the least tension in the spring and the maximum acceleration of \(Q\) during the motion. [5]

OR A shop is supplied with large quantities of plant pots in packs of six. These pots can be damaged easily if they are not packed carefully. The manager of the shop is a statistician and he believes that the number of damaged pots in a pack of six has a binomial distribution. He chooses a random sample of 250 packs and records the numbers of damaged pots per pack. His results are shown in the following table.

1. Show that the mean number of damaged pots per pack in this sample is 1.656. [1]

The following table shows some of the expected frequencies, correct to 2 decimal places, using an appropriate binomial distribution.

1. Find the values of \(a\) and \(b\), correct to 2 decimal places [5]
2. Use a goodness-of-fit test at the 1% significance level to determine whether the manager's belief is justified. [8]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(3m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(O\) on a smooth plane that is inclined to the horizontal at an angle \(\alpha\), where \(\sin\alpha = \frac{3}{5}\). The system rests in equilibrium with \(P\) on the plane at the point \(E\). The length of the spring in this position is \(\frac{5}{4}a\).

1. Find the value of \(k\). [3]
2. The particle \(P\) is now replaced by a particle \(Q\) of mass \(2m\) and \(Q\) is released from rest at the point \(E\). Show that, in the resulting motion, \(Q\) performs simple harmonic motion. State the centre and the period of the motion. [6]
3. Find the least tension in the spring and the maximum acceleration of \(Q\) during the motion. [5]

OR A shop is supplied with large quantities of plant pots in packs of six. These pots can be damaged easily if they are not packed carefully. The manager of the shop is a statistician and he believes that the number of damaged pots in a pack of six has a binomial distribution. He chooses a random sample of 250 packs and records the numbers of damaged pots per pack. His results are shown in the following table.

Number of damaged pots per pack \((x)\)	0	1	2	3	4	5	6
Frequency	48	69	78	32	22	1	0

1. Show that the mean number of damaged pots per pack in this sample is 1.656. [1]
2. The following table shows some of the expected frequencies, correct to 2 decimal places, using an appropriate binomial distribution.

   Number of damaged pots per pack \((x)\)	0	1	2	3	4	5	6
   Expected frequency	36.01	82.36	\(a\)	39.89	\(b\)	1.74	0.11

   Find the values of \(a\) and \(b\), correct to 2 decimal places [5]
3. Use a goodness-of-fit test at the 1% significance level to determine whether the manager's belief is justified. [8]

A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\), and the amplitude of the motion is \(2.5\) m. The points \(L\) and \(M\) are on the line, on opposite sides of \(O\), with \(OL = 1.5\) m. The magnitudes of the accelerations of \(P\) at \(L\) and at \(M\) are in the ratio \(3 : 4\).

1. Find the distance \(OM\). [2]

The time taken by \(P\) to travel directly from \(L\) to \(M\) is \(2\) s.

1. Find the period of the motion. [5]
2. Find the speed of \(P\) when it passes through \(L\). [2]

A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line on opposite sides of \(O\) such that \(OA = 3.5\) m and \(OB = 1\) m. The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is 1 m s\(^{-2}\).

1. Find the speed of \(P\) when it is at \(O\). [4]
2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\). [4]

\includegraphics{figure_5} A thin uniform rod \(AB\) has mass \(kM\) and length \(2a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(kM\) and radius \(2a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C\), \(B\), \(A\), \(O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).

1. Show that the moment of inertia of the object about \(L\) is \(\frac{5}{2}(8k + 3)Ma^2\). [6] The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
2. Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\). [6]

\(A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(AB = 3a\) m. One end of a light elastic string, of natural length \(a\) m and modulus of elasticity \(mg\) N, is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m\) kg. One end of a second light elastic string, of natural length \(ka\) m and modulus of elasticity \(2mg\) N, is attached to \(B\). The other end of this string is attached to \(P\). Given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(AB\), find the value of \(k\). [3] 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. [5] In the case where \(P\) is released from rest at a distance \(0.2a\) m from \(M\), the speed of \(P\) is \(0.7\) m s\(^{-1}\) when \(P\) is \(0.05a\) m from \(M\). Find the value of \(a\). [3]

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]

The fixed points \(A\) and \(B\) are on a smooth horizontal surface with \(AB = 2.6\) m. One end of a light elastic spring, of natural length 1.25 m and modulus of elasticity \(0.6\) N, is attached to \(A\). The other end is attached to a particle \(P\) of mass 0.4 kg. One end of a second light elastic spring, of natural length 1.0 m and modulus of elasticity \(0.62\) N, is attached to \(B\); its other end is attached to \(P\). The system is in equilibrium with \(P\) on the surface at the point \(E\).

1. Show that \(AE = 1.4\) m. [4]

The particle \(P\) is now displaced slightly from \(E\), along the line \(AB\).

1. Show that, in the subsequent motion, \(P\) performs simple harmonic motion. [5]
2. Given that the period of the motion is \(\frac{4}{\pi}\) s, find the value of \(\lambda\). [3]

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]

An object is formed from a uniform circular disc, of radius \(2a\) and mass \(3M\), and a uniform rod \(AB\), of length \(4a\) and mass \(kM\), where \(k\) is a constant. The centre of the disc is \(O\). The end \(B\) of the rod is rigidly joined to a point on the circumference of the disc so that \(OBA\) is a straight line. The fixed horizontal axis \(l\) is in the plane of the object, passes through \(A\) and is perpendicular to \(AB\).

1. Show that the moment of inertia of the object about the axis \(l\) is \(3Ma^2(26 + k)\). [5]
2. The object is free to rotate about \(l\). Show that small oscillations of the object about \(l\) are approximately simple harmonic. Given that the period of these oscillations is \(4\pi\sqrt{\frac{a}{g}}\), find the value of \(k\). [6]

Answer only one of the following two alternatives. **EITHER** One end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 40 N, is attached to a fixed point \(O\). The spring hangs vertically, at rest, with particles of masses 2 kg and \(M\) kg attached to its free end. The \(M\) kg particle becomes detached from the spring, and as a result the 2 kg particle begins to move upwards. \begin{enumerate}[label=(\roman*)] \item Show that the 2 kg particle performs simple harmonic motion about its equilibrium position with period \(\frac{2\pi}{5}\) s. State the distance below \(O\) of the centre of the oscillations. [7] \item The speed of the 2 kg particle is 0.4 m s\(^{-1}\) when its displacement from the centre of oscillation is 0.06 m. Find the amplitude of the motion. [3] \item Deduce the value of \(M\). [4] \end{enumerate] **OR** In a particular country, large numbers of ducks live on lakes \(A\) and \(B\). The mass, in kg, of a duck on lake \(A\) is denoted by \(x\) and the mass, in kg, of a duck on lake \(B\) is denoted by \(y\). A random sample of 8 ducks is taken from lake \(A\) and a random sample of 10 ducks is taken from lake \(B\). Their masses are summarised as follows. \(\Sigma x = 10.56\) \(\quad\) \(\Sigma x^2 = 14.1775\) \(\quad\) \(\Sigma y = 12.39\) \(\quad\) \(\Sigma y^2 = 15.894\) A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\). \begin{enumerate}[label=(\roman*)] \item Test, at the 10% significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal. [9] \item A second random sample of 8 ducks is taken from lake \(A\) and their masses are summarised as \(\Sigma x = 10.24\) \(\quad\) and \(\quad\) \(\Sigma(x - \bar{x})^2 = 0.294\), where \(\bar{x}\) is the sample mean. The scientist now claims that the population mean mass of ducks on lake \(A\) is greater than \(p\) kg. A test of this claim is carried out at the 10% significance level, using only this second sample from lake \(A\). This test supports the scientist's claim. Find the greatest possible value of \(p\). [5] \end{enumerate]

\includegraphics{figure_5} A thin uniform rod \(AB\) has mass \(\lambda M\) and length \(2a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(CBAO\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac{1}{2}a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).

1. Show that the moment of inertia of the object about \(L\) is \(\left(\frac{408 + 7\lambda}{12}\right)Ma^2\). [6]

The period of small oscillations of the object about \(L\) is \(5\pi\sqrt{\left(\frac{2a}{g}\right)}\).

1. Find the value of \(\lambda\). [6]

Answer only one of the following two alternatives. EITHER The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac{2}{3}\) kg is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N. The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N, has one end attached to \(P\) and the other end attached to \(B\).

1. Show that when \(P\) is in equilibrium \(AP = 0.75\) m. [3]

The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.

1. Show that \(P\) performs simple harmonic motion and state the period of the motion. [6]
2. Find the speed of \(P\) when it passes through the equilibrium position. [2]
3. Find the speed of \(P\) when its acceleration is equal to half of its maximum value. [3]

OR The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.

Number of repairs in a week	0	1	2	3	4	5	\(\geqslant 6\)
Number of weeks	6	15	9	6	3	1	0

1. Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data. [3]

Records over a longer period of time indicate that the mean number of repairs in a week is 1.6. The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.

Number of repairs in a week	0	1	2	3	4	5	\(\geqslant 6\)
Expected frequency	8.076	12.921	10.337	5.513	2.205	\(a\)	\(b\)

1. Show that \(a = 0.706\) and find the value of the constant \(b\). [3]
2. Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a 10% significance level. [8]

$$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 6x = 2e^{-t}.$$ Given that \(x = 0\) and \(\frac{dx}{dt} = 2\) at \(t = 0\),

1. find \(x\) in terms of \(t\). [8]

The solution to part (a) is used to represent the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds, where \(t > 0\), \(P\) is \(x\) metres from the origin \(O\).

1. Show that the maximum distance between \(O\) and \(P\) is \(\frac{2\sqrt{3}}{9}\) m and justify that this distance is a maximum. [7]

A light elastic spring, of natural length \(5a\) and modulus of elasticity \(10mg\), has one end attached to a fixed point \(A\) on a ceiling. A particle \(P\) of mass \(2m\) is attached to the other end of the spring and \(P\) hangs freely in equilibrium at the point \(O\).

1. Find the distance \(AO\). [3]

The particle is now pulled vertically downwards a distance \(\frac{1}{2}a\) from \(O\) and released from rest.

1. Show that \(P\) moves with simple harmonic motion. [4]
2. Find the period of the motion. [2]

A vertical ladder is fixed to a wall in a harbour. On a particular day the minimum depth of water in the harbour occurs at 0900 hours. The next time the water is at its minimum depth is 2115 hours on the same day. The bottom step of the ladder is 1 m above the lowest level of the water and 9 m below the highest level of the water. The rise and fall of the water level can be modelled as simple harmonic motion and the thickness of the step can be assumed to be negligible. Find

1. the speed, in metres per hour, at which the water level is moving when it reaches the bottom step of the ladder, [7]
2. the length of time, on this day, between the water reaching the bottom step of the ladder and the ladder being totally out of the water once more. [4]

A light elastic string, of natural length \(3a\) and modulus of elasticity \(6mg\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(2m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\), vertically below \(A\).

1. Find the distance \(AO\). [3]

The particle is now raised to point \(C\) vertically below \(A\), where \(AC > 3a\), and is released from rest.

1. Show that \(P\) moves with simple harmonic motion of period \(2\pi\sqrt{\frac{a}{g}}\). [5]

It is given that \(OC = \frac{1}{4}a\).

1. Find the greatest speed of \(P\) during the motion. [3]

The point \(D\) is vertically above \(O\) and \(OD = \frac{1}{8}a\). The string is cut as \(P\) passes through \(D\), moving upwards.

1. Find the greatest height of \(P\) above \(O\) in the subsequent motion. [4]

The points \(O\), \(A\), \(B\) and \(C\) lie in a straight line, in that order, where \(OA = 0.6\) m, \(OB = 0.8\) m and \(OC = 1.2\) m. A particle \(P\), moving along this straight line, has a speed of \(\left(\frac{1}{10}\sqrt{5}\right)\) m s\(^{-1}\) at \(A\), \(\left(\frac{1}{5}\sqrt{5}\right)\) m s\(^{-1}\) at \(B\) and is instantaneously at rest at \(C\).

1. Show that this information is consistent with \(P\) performing simple harmonic motion with centre \(O\). [5]

Given that \(P\) is performing simple harmonic motion with centre \(O\),

1. show that the speed of \(P\) at \(O\) is 0.6 m s\(^{-1}\), [2]
2. find the magnitude of the acceleration of \(P\) as it passes \(A\), [2]
3. find, to 3 significant figures, the time taken for \(P\) to move directly from \(A\) to \(B\). [4]

In a game at a fair, a small target \(C\) moves horizontally with simple harmonic motion between the points \(A\) and \(B\), where \(AB = 4L\). The target moves inside a box and takes 3 s to travel from \(A\) to \(B\). A player has to shoot at \(C\), but \(C\) is only visible to the player when it passes a window \(PQ\), where \(PQ = b\). The window is initially placed with \(Q\) at the point as shown in Figure 4. The target \(C\) takes 0.75 s to pass from \(Q\) to \(P\).

1. Show that \(b = (2 - \sqrt{2})L\). [5]
2. Find the speed of \(C\) as it passes \(P\). [2]

\includegraphics{figure_5} For advanced players, the window \(PQ\) is moved to the centre of \(AB\) so that \(AP = QB\), as shown in Figure 5.

1. Find the time, in seconds to 2 decimal places, taken for \(C\) to pass from \(Q\) to \(P\) in this new position. [3]

A light spring of natural length \(L\) has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The particle is moving vertically. As it passes through the point \(B\) below \(A\), where \(AB = L\), its speed is \(\sqrt{(2gL)}\). The particle comes to instantaneous rest at a point \(C\), \(4L\) below \(A\).

1. Show that the modulus of elasticity of the spring is \(\frac{8mg}{9}\). [4]

At the point \(D\) the tension in the spring is \(mg\).

1. Show that \(P\) performs simple harmonic motion with centre \(D\). [5]
2. Find, in terms of \(L\) and \(g\),
   1. the period of the simple harmonic motion,
   2. the maximum speed of \(P\).
   [5]

A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds its displacement, \(x\) metres, from the origin \(O\) is given by \(x = 5 \sin (\frac{1}{4}\pi t)\).

1. Prove that \(P\) is moving with simple harmonic motion. [3]
2. Find the period and the amplitude of the motion. [2]
3. Find the maximum speed of \(P\). [2]

The points \(A\) and \(B\) on the positive \(x\)-axis are such that \(OA = 2\) m and \(OB = 3\) m.

1. Find the time taken by \(P\) to travel directly from \(A\) to \(B\). [4]

\includegraphics{figure_5} A small ring \(R\) of mass \(m\) is free to slide on a smooth straight wire which is fixed at an angle of \(30°\) to the horizontal. The ring is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point \(B\), where \(AB = \frac{a}{2}\).

1. Show that \(\lambda = 4mg\). [3]

The ring is pulled down to the point \(C\), where \(BC = \frac{1}{4}a\), and released from rest. At time \(t\) after \(R\) is released the extension of the string is \((\frac{1}{4}a + x)\).

1. Obtain a differential equation for the motion of \(R\) while the string remains taut, and show that it represents simple harmonic motion with period \(\pi\sqrt{\left(\frac{a}{g}\right)}\). [6]
2. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(R\) while the string remains taut. [2]
3. Find, in terms of \(a\) and \(g\), the time taken for \(R\) to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. [5]

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]