Equation of motion angular acceleration

2
\includegraphics[max width=\textwidth, alt={}, center]{34024618-0ff9-44a1-ac57-d4d7e8a3655e-2_431_421_881_861} A uniform disc of radius 0.4 m is free to rotate without friction in a vertical plane about a horizontal axis through its centre. The moment of inertia of the disc about the axis is \(0.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a particle of mass 1.5 kg which hangs freely (see diagram). The system is released from rest. Find

1. the angular acceleration of the disc,
2. the speed of the particle when the disc has turned through an angle of \(\frac { 1 } { 6 } \pi\).

2
\includegraphics[max width=\textwidth, alt={}, center]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-2_431_421_881_861} A uniform disc of radius 0.4 m is free to rotate without friction in a vertical plane about a horizontal axis through its centre. The moment of inertia of the disc about the axis is \(0.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a particle of mass 1.5 kg which hangs freely (see diagram). The system is released from rest. Find

1. the angular acceleration of the disc,
2. the speed of the particle when the disc has turned through an angle of \(\frac { 1 } { 6 } \pi\).

4 A uniform circular disc has mass \(m\), radius \(a\) and centre \(C\). The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point \(A\) on the disc, where \(C A = \frac { 1 } { 3 } a\).

1. Find the moment of inertia of the disc about this axis. The disc is released from rest with \(C A\) horizontal.
2. Find the initial angular acceleration of the disc.
3. State the direction of the force acting on the disc at \(A\) immediately after release, and find its magnitude.

1. A body consists of a uniform plane circular disc, of radius \(r\) and mass \(2 m\), with a particle of mass \(3 m\) attached to the circumference of the disc at the point \(P\).
   The line \(P Q\) is a diameter of the disc. The body is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), which is perpendicular to the plane of the disc and passes through \(Q\). The body is held with \(Q P\) making an angle \(\beta\) with the downward vertical through \(Q\), where \(\sin \beta = 0.25\), and released from rest. Find the magnitude of the component, perpendicular to \(P Q\), of the force acting on the body at \(Q\) at the instant when it is released.
   [0pt] [You may assume that the moment of inertia of the body about \(L\) is \(15 m r ^ { 2 }\).]
2. The points \(P\) and \(Q\) have position vectors \(4 \mathbf { i } - 6 \mathbf { j } - 12 \mathbf { k }\) and \(2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k }\) respectively, relative to a fixed origin \(O\).

Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\), act along \(\overrightarrow { O P } , \overrightarrow { Q O }\) and \(\overrightarrow { Q P }\) respectively, and have magnitudes \(7 \mathrm {~N} , 3 \mathrm {~N}\) and \(3 \sqrt { } 10 \mathrm {~N}\) respectively.

1. Express \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) in vector form.
2. Show that the resultant of \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) is \(( 2 \mathbf { i } - 10 \mathbf { j } - 16 \mathbf { k } ) \mathrm { N }\).
3. Find a vector equation of the line of action of this resultant, giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a parameter.