Small oscillations: simple pendulum (particle on string)

7 A simple pendulum consists of a light inextensible string of length 0.8 m and a particle \(P\) of mass \(m \mathrm {~kg}\). The pendulum is hanging vertically at rest from a fixed point \(O\) when \(P\) is given a horizontal velocity of \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, in the subsequent motion, the maximum angle between the string and the downward vertical is 0.107 radians, correct to 3 significant figures.
2. Show that the motion may be modelled as simple harmonic motion, and find the period of this motion.
3. Find the time after the start of the motion when the velocity of the particle is first \(- 0.2 \mathrm {~ms} ^ { - 1 }\) and find the angular displacement of \(O P\) from the downward vertical at this time.

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.

10 A particle \(P\) is attached to one end of a light inextensible string of length 1.4 m . The other end of the string is fixed to the ceiling at \(C\). The angle between \(C P\) and the vertical is \(\theta\) radians. The particle is held with the string taut with \(\theta = 0.3\) and is then released.

1. (a) Show that the motion of the system is approximately simple harmonic, and state its period.
   (b) Hence find an approximation for the speed of \(P\) when \(\theta = 0.2\).
2. Find the speed of \(P\) when \(\theta = 0.2\) using an energy method, and hence find the percentage error in the answer to part (i) (b).

6 A simple pendulum consists of a light inextensible string of length 1.5 m with a small bob of mass 0.2 kg at one end. When suspended from a fixed point and hanging at rest under gravity, the bob is given a horizontal speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it comes instantaneously to rest when the string makes an angle of 0.1 rad with the vertical. At time \(t\) seconds after projection the string makes an angle \(\theta\) with the vertical.

1. Show that, neglecting air resistance, $$\left( \frac { \mathrm { d } \theta } { \mathrm {~d} t } \right) ^ { 2 } = \frac { 40 } { 3 } \{ \cos \theta - \cos ( 0.1 ) \}$$
2. Find, correct to 2 significant figures,
   1. the value of \(u\),
   2. the tension in the string when \(\theta = 0.05 \mathrm { rad }\).
   3. By differentiating the above equation for \(\left( \frac { \mathrm { d } \theta } { \mathrm { d } t } \right) ^ { 2 }\), or otherwise, show that the motion of the bob can be modelled approximately by simple harmonic motion.
   4. Hence find the value of \(t\) at which the bob first comes instantaneously to rest.

\includegraphics{figure_6} A particle \(P\) of mass \(m\) kg is attached to one end of a light inextensible string of length \(L\) m. The other end of the string is attached to a fixed point \(O\). The particle is held at rest with the string taut and then released. \(P\) starts to move and in the subsequent motion the angular displacement of \(OP\), at time \(t\) s, is \(\theta\) radians from the downward vertical (see diagram). The initial value of \(\theta\) is \(0.05\).

1. Show that \(\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin \theta\). [2]
2. Hence show that the motion of \(P\) is approximately simple harmonic. [2]
3. Given that the period of the approximate simple harmonic motion is \(\frac{4}{3}\pi\) s, find the value of \(L\). [2]
4. Find the value of \(\theta\) when \(t = 0.7\) s, and the value of \(t\) when \(\theta\) next takes this value. [4]
5. Find the speed of \(P\) when \(t = 0.7\) s. [3]

One end of a light inextensible string of length \(2\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.2\) kg is attached to the other end of the string. \(P\) is held at rest with the string taut so that \(OP\) makes an angle of \(0.15\) radians with the downward vertical. \(P\) is released and \(t\) seconds afterwards \(OP\) makes an angle of \(\theta\) radians with the downward vertical.

1. Show that \(\frac{d^2\theta}{dt^2} = -4.9 \sin \theta\) and give a reason why the motion is approximately simple harmonic. [3]

Using the simple harmonic approximation,

1. obtain an expression for \(\theta\) in terms of \(t\) and hence find the values of \(t\) at the first and second occasions when \(\theta = -0.1\), [5]
2. find the angular speed of \(OP\) and the linear speed of \(P\) when \(t = 0.5\). [3]