Maximum speed in SHM

2 The tip of a sewing-machine needle oscillates vertically in simple harmonic motion through a distance of 2.10 cm . It takes 2.25 s to perform 100 complete oscillations. Find, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), the maximum speed of the tip of the needle. Show that the speed of the tip when it is at a distance of 0.5 cm from a position of instantaneous rest is \(2.50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.

1 A particle \(P\) is moving with simple harmonic motion in a straight line. The period is 6.1 s and the amplitude is 3 m . Calculate, in either order,

1. the maximum speed of \(P\),
2. the distance of \(P\) from the centre of motion when \(P\) has speed \(2.5 \mathrm {~ms} ^ { - 1 }\).

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. The diagram below shows part of a game at a funfair that consists of a target moving along a straight horizontal line \(A B\). The centre of the target may be modelled as a particle moving with Simple Harmonic Motion about centre \(O\), where \(O\) is the midpoint of \(A B\). \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-14_245_1145_525_452}

When the target is at a distance of 84 cm from \(O\), its speed is \(52 \mathrm { cms } ^ { - 1 }\) and the magnitude of its acceleration is \(1344 \mathrm { cms } ^ { - 2 }\).

1. Show that the period of the motion is \(\frac { \pi } { 2 } \mathrm {~s}\).
   

   (b) Determine the maximum speed of the target.

   Examiner

2. During a game, players fire a ball at the target. A timer is started when the target is at \(A\). Players must wait for the target to complete at least one full cycle before firing. Given that the target is hit when it is at a distance of 67 cm from \(O\), calculate the two earliest possible times taken to hit the target.
   \section*{PLEASE DO NOT WRITE ON THIS PAGE}

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 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]

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]

\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 light elastic string, of natural length \(4a\) and modulus of elasticity \(8mg\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\).

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

The particle is now pulled down to a point \(C\) vertically below \(O\), where \(OC = d\). It is released from rest. In the subsequent motion the string does not become slack.

1. Show that \(P\) moves with simple harmonic motion of period \(\pi\sqrt{\frac{2a}{g}}\). [7]

The greatest speed of \(P\) during this motion is \(\frac{1}{2}\sqrt{(ga)}\).

1. Find \(d\) in terms of \(a\). [3]

Instead of being pulled down a distance \(d\), the particle is pulled down a distance \(a\). Without further calculation,

1. describe briefly the subsequent motion of \(P\). [2]

A particle \(B\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.75 m and modulus of elasticity 24.5 N. The other end of the string is attached to a fixed point \(A\). The particle is hanging in equilibrium at the point \(E\), vertically below \(A\).

1. Show that \(AE = 0.9\) m. [3]

The particle is held at \(A\) and released from rest. The particle first comes to instantaneous rest at the point \(C\).

1. Find the distance \(AC\). [5]
2. Show that while the string is taut, \(B\) is moving with simple harmonic motion with centre \(E\). [4]
3. Calculate the maximum speed of \(B\). [2]

A light elastic string, of natural length \(l\) m and modulus of elasticity \(\frac{mg}{2}\) newtons, has one end fastened to a fixed point \(O\). A particle \(P\), of mass \(m\) kg, is attached to the other end of the string. \(P\) hangs in equilibrium at the point \(E\), vertically below \(O\), where \(OE = (l + e)\) m

1. Find the numerical value of the ratio \(e : l\). [2 marks]

\(P\) is now pulled down a further distance \(\frac{3l}{2}\) m from \(E\) and is released from rest. In the subsequent motion, the string remains taut. At time \(t\) s after being released, \(P\) is at a distance \(x\) m below \(E\).

1. Write down a differential equation for the motion of \(P\) and show that the motion is simple harmonic. [4 marks]
2. Write down the period of the motion. [2 marks]
3. Find the speed with which \(P\) first passes through \(E\) again. [2 marks]
4. Show that the time taken by \(P\) after it is released to reach the point \(A\) above \(E\), where \(AE = \frac{3l}{4}\) m, is \(\frac{2\pi}{3}\sqrt{\frac{2l}{g}}\) s. [5 marks]

A light elastic string, of natural length 0·8 m, has one end fastened to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0·5 kg. When \(P\) hangs in equilibrium, the length of the string is 1·5 m.

1. Calculate the modulus of elasticity of the string. [3 marks]

\(P\) is displaced to a point 0·5 m vertically below its equilibrium position and released from rest. \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Show that the subsequent motion of \(P\) is simple harmonic, with period 1·68 s. [5 marks] \item Calculate the maximum speed of \(P\) during its motion. [3 marks] \item Show that the time taken for \(P\) to first reach a distance 0·25 m from the point of release is 0·28 s, to 2 significant figures. [4 marks] \end{enumerate]

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\(^{-1}\)

One end of a light spring of length 0.5 m is attached to a fixed point \(F\). A particle \(P\) of mass 2.5 kg is attached to the other end of the spring and hangs in equilibrium 0.55 m below \(F\). Another particle \(Q\), of mass 1.5 kg, is attached to \(P\), without moving it, and both particles are then released.

1. Show that the modulus of elasticity of the spring is 250 N. [2]
2. Prove that the motion is simple harmonic. [4]
3. Find
   1. the amplitude of the motion, [1]
   2. the greatest speed of the particles, [1]
   3. the period of the motion, [1]
   4. the time taken for the spring to be extended by 0.1 m for the first time. [3]