Angular speed and period

1 The end \(A\) of a \(\operatorname { rod } A B\) of length 1.2 m is freely pivoted at a fixed point. The rod rotates about \(A\) in a vertical plane. Calculate the angular speed of the rod at an instant when \(B\) has speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1 A particle \(P\) is moving in a circle of radius 1.5 m . At time \(t \mathrm {~s}\) its velocity is \(\left( k - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a positive constant. When \(t = 3\), the magnitudes of the radial and transverse components of the acceleration of \(P\) are equal. Find the possible values of \(k\).

A uniform rectangular lamina \(ABCD\) has mass 20 kg and sides of lengths \(AB = 0.6\) m and \(BC = 1.8\) m. It rotates in its own vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the mid-point of \(AB\).

1. Show that the moment of inertia of the lamina about the axis is 22.2 kg m\(^2\). [3]

\includegraphics{figure_5} The lamina is released from rest with \(BC\) horizontal and below the level of the axis. Air resistance may be neglected, but a frictional couple opposes the motion. The couple has constant moment 44.1 N m about the axis. The angle through which the lamina has turned is denoted by \(\theta\) (see diagram).

1. Show that the angular acceleration is zero when \(\cos \theta = 0.25\). [3]
2. Hence find the maximum angular speed of the lamina. [5]

\includegraphics{figure_7} A uniform rod \(AB\) has mass \(m\) and length \(6a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(C\) on the rod, where \(AC = a\). The angle between \(AB\) and the upward vertical is \(\theta\), and the force acting on the rod at \(C\) has components \(R\) parallel to \(AB\) and \(S\) perpendicular to \(AB\) (see diagram). The rod is released from rest in the position where \(\theta = \frac{1}{4}\pi\). Air resistance may be neglected.

1. Find the angular acceleration of the rod in terms of \(a\), \(g\) and \(\theta\). [4]
2. Show that the angular speed of the rod is \(\sqrt{\frac{2g(1 - 2\cos\theta)}{7a}}\). [3]
3. Find \(R\) and \(S\) in terms of \(m\), \(g\) and \(\theta\). [6]
4. When \(\cos\theta = \frac{1}{3}\), show that the force acting on the rod at \(C\) is vertical, and find its magnitude. [4]

A uniform square lamina, of mass 5 kg and side 0.2 m, is rotating about a fixed vertical axis that is perpendicular to the lamina and that passes through its centre. A couple of constant moment 0.06 N m is applied to the lamina. The lamina turns through an angle of 155 radians while its angular speed increases from 8 rad s\(^{-1}\) to \(\omega\) rad s\(^{-1}\). Find \(\omega\). [4]

\includegraphics{figure_6} A compound pendulum consists of a uniform rod \(AB\) of length 1 m and mass 3 kg, a particle of mass 1 kg attached to the rod at \(A\) and a circular disc of radius \(\frac{1}{5}\) m, mass 6 kg and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the circumference of the disc in such a way that \(ABC\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(P\) on the rod where \(AP = x\) m and \(x < \frac{1}{3}\) (see diagram).

1. Show that the moment of inertia of the pendulum about the axis of rotation is \((10x^2 - 19x + 12)\) kg m\(^2\). [6]

The pendulum is making small oscillations about the equilibrium position, such that at time \(t\) seconds the angular displacement that the pendulum makes with the downward vertical is \(\theta\) radians.

1. Find the angular acceleration of the pendulum, in terms of \(x\), \(g\) and \(\theta\). [4]
2. Show that the motion is approximately simple harmonic, and show that the approximate period of oscillations, in seconds, is given by \(2\pi\sqrt{\frac{20x^2 - 38x + 24}{(19-20x)g}}\). [2]
3. Hence find the value of \(x\) for which the approximate period of oscillations is least. [3]

A pendulum consists of a uniform rod \(AB\), of length \(4a\) and mass \(2m\), whose end \(A\) is rigidly attached to the centre \(O\) of a uniform square lamina \(PQRS\), of mass \(4m\) and side \(a\). The rod \(AB\) is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(B\). The axis \(L\) is perpendicular to \(AB\) and parallel to the edge \(PQ\) of the square.

1. Show that the moment of inertia of the pendulum about \(L\) is \(75ma^2\). [4]

The pendulum is released from rest when \(BA\) makes an angle \(\alpha\) with the downward vertical through \(B\), where \(\tan \alpha = \frac{3}{4}\). When \(BA\) makes an angle \(\theta\) with the downward vertical through \(B\), the magnitude of the component, in the direction \(AB\), of the force exerted by the axis \(L\) on the pendulum is \(X\).

1. Find an expression for \(X\) in terms of \(m\), \(g\) and \(\theta\). [9]

Using the approximation \(\theta \approx \sin \theta\),

1. find an estimate of the time for the pendulum to rotate through an angle \(\alpha\) from its initial rest position. [6]

\includegraphics{figure_2} **Figure 1** A uniform circular disc has mass \(4m\), centre \(O\) and radius \(4a\). The line \(POQ\) is a diameter of the disc. A circular hole of radius \(2a\) is made in the disc with the centre of the hole at the point \(R\) on \(PQ\) where \(QR = 5a\), as shown in Figure 1. The resulting lamina is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(Q\) and is perpendicular to the plane of the lamina.

1. Show that the moment of inertia of the lamina about \(L\) is \(69ma^2\). [7]

The lamina is hanging at rest with \(P\) vertically below \(Q\) when it is given an angular velocity \(\Omega\). Given that the lamina turns through an angle \(\frac{2\pi}{3}\) before it first comes to instantaneous rest,

1. find \(\Omega\) in terms of \(g\) and \(a\). [6]

A uniform lamina \(ABC\) of mass \(m\) is in the shape of an isosceles triangle with \(AB = AC = 5a\) and \(BC = 8a\).

1. Show, using integration, that the moment of inertia of the lamina about an axis through \(A\), parallel to \(BC\), is \(\frac{9}{2}ma^2\). [6]

The foot of the perpendicular from \(A\) to \(BC\) is \(D\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through \(D\) and is perpendicular to the plane of the lamina. The lamina is released from rest when \(DA\) makes an angle \(\alpha\) with the downward vertical. It is given that the moment of inertia of the lamina about an axis through \(D\), perpendicular to \(BC\) and in the plane of the lamina, is \(\frac{8}{3}ma^2\).

1. Find the angular acceleration of the lamina when \(DA\) makes an angle \(\theta\) with the downward vertical. [8]

Given that \(\alpha\) is small,

1. find an approximate value for the period of oscillation of the lamina about the vertical. [2]

A pendulum consists of a uniform rod \(PQ\), of mass \(3m\) and length \(2a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.

1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac{33}{4}ma^2\). [5]

The pendulum is released from rest in the position where \(PQ\) makes an angle \(\alpha\) with the downward vertical. At time \(t\), \(PQ\) makes an angle \(\theta\) with the downward vertical.

1. Show that the angular speed, \(\dot{\theta}\), of the pendulum satisfies $$\dot{\theta}^2 = \frac{40g(\cos \theta - \cos \alpha)}{33a}$$ [4]

1. Hence, or otherwise, find the angular acceleration of the pendulum. [3]

Given that \(\alpha = \frac{\pi}{20}\) and that \(PQ\) has length \(\frac{8}{33}\) m,

1. find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest. [5]

A uniform circular disc, of mass \(m\), radius \(a\) and centre \(O\), is free to rotate in a vertical plane about a fixed smooth horizontal axis. The axis passes through the mid-point \(A\) of a radius of the disc.

1. Find an equation of motion for the disc when the line \(AO\) makes an angle \(\theta\) with the downward vertical through \(A\). [5]
2. Hence find the period of small oscillations of the disc about its position of stable equilibrium. [2]

When the line \(AO\) makes an angle \(\theta\) with the downward vertical through \(A\), the force acting on the disc at \(A\) is \(\mathbf{F}\).

1. Find the magnitude of the component of \(\mathbf{F}\) perpendicular to \(AO\). [5]

A pendulum consists of a uniform rod \(PQ\), of mass \(3m\) and length \(2a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.

1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac{33}{4}ma^2\). [5]

The pendulum is released from rest in the position where \(PQ\) makes an angle \(\alpha\) with the downward vertical. At time \(t\), \(PQ\) makes an angle \(\theta\) with the downward vertical.

1. Show that the angular speed, \(\dot{\theta}\), of the pendulum satisfies $$\dot{\theta}^2 = \frac{40g(\cos\theta - \cos\alpha)}{33a}$$ [4]
2. Hence, or otherwise, find the angular acceleration of the pendulum. [3]

Given that \(\alpha = \frac{\pi}{20}\) and that \(PQ\) has length \(\frac{8}{33}\) m,

1. find, to 3 significant figures, an approximate value for the angular speed of the pendulum \(0.2\) s after it has been released from rest. [5]