OCR M4 (Mechanics 4) 2011 June

1 When the power is turned off, a fan disk inside a jet engine slows down with constant angular deceleration \(0.8 \mathrm { rad } \mathrm { s } ^ { - 2 }\).

1. Find the time taken for the angular speed to decrease from \(950 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(750 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Find the angle through which the disk turns as the angular speed decreases from \(220 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(200 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
3. Find the time taken for the disk to make the final 10 revolutions before coming to rest.

2 A straight \(\operatorname { rod } A B\) has length \(a\). The rod has variable density, and at a distance \(x\) from \(A\) its mass per unit length is \(k \mathrm { e } ^ { - \frac { x } { a } }\), where \(k\) is a constant. Find, in an exact form, the distance of the centre of mass of the rod from \(A\).

3 A uniform rod \(X Y\), of mass 5 kg and length 1.8 m , is free to rotate in a vertical plane about a fixed horizontal axis through \(X\). The rod is at rest with \(Y\) vertically below \(X\) when a couple of constant moment is applied to the rod. It then rotates, and comes instantaneously to rest when \(X Y\) is horizontal.

1. Find the moment of the couple.
2. Find the angular acceleration of the rod
   (a) immediately after the couple is first applied,
   (b) when \(X Y\) is horizontal.

4
\includegraphics[max width=\textwidth, alt={}, center]{337dd1f9-a691-4e99-9aa7-7a93d8bb13be-2_439_1045_1512_550} Two small smooth pegs \(A\) and \(B\) are fixed at a distance \(2 a\) apart on the same horizontal level, and \(C\) is the mid-point of \(A B\). A uniform rod \(C D\), of mass \(m\) and length \(a\), is freely pivoted at \(C\) and can rotate in the vertical plane containing \(A B\), with \(D\) below the level of \(A B\). A light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\), passes round the peg \(A\) and its ends are attached to \(C\) and \(D\). Another light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), passes round the peg \(B\) and its ends are also attached to \(C\) and \(D\). The angle \(C A D\) is \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), so that the angle \(B C D\) is \(2 \theta\) (see diagram).

1. Taking \(A B\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$\frac { 1 } { 2 } m g a ( 14 - 2 \cos 2 \theta - \sin 2 \theta )$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine whether this position of equilibrium is stable or unstable.

5 The region inside the circle \(x ^ { 2 } + y ^ { 2 } = a ^ { 2 }\) is rotated about the \(x\)-axis to form a uniform solid sphere of radius \(a\) and volume \(\frac { 4 } { 3 } \pi a ^ { 3 }\). The mass of the sphere is \(10 M\).

1. Show by integration that the moment of inertia of the sphere about the \(x\)-axis is \(4 M a ^ { 2 }\). (You may assume the standard formula \(\frac { 1 } { 2 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about its axis.) The sphere is free to rotate about a fixed horizontal axis which is a diameter of the sphere. A particle of mass \(M\) is attached to the lowest point of the sphere. The sphere with the particle attached then makes small oscillations as a compound pendulum.
2. Find, in terms of \(a\) and \(g\), the approximate period of these oscillations.

6 Two ships \(P\) and \(Q\) are moving on straight courses with constant speeds. At one instant \(Q\) is 80 km from \(P\) on a bearing of \(220 ^ { \circ }\). Three hours later, \(Q\) is 36 km due south of \(P\).

1. Show that the velocity of \(Q\) relative to \(P\) is \(19.1 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(063.8 ^ { \circ }\) (both correct to 3 significant figures).
2. Find the shortest distance between the two ships in the subsequent motion. Given that the speed of \(P\) is \(28 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and \(Q\) is travelling in the direction with bearing \(105 ^ { \circ }\), find
3. the bearing of the direction in which \(P\) is travelling,
4. the speed of \(Q\).

7
\includegraphics[max width=\textwidth, alt={}, center]{337dd1f9-a691-4e99-9aa7-7a93d8bb13be-3_479_1225_1484_461} A uniform rectangular block of mass \(m\) and cross-section \(A B C D\) has \(A B = C D = 6 a\) and \(A D = B C = 2 a\). The point \(X\) is on \(A B\) such that \(A X = a\) and \(G\) is the centre of \(A B C D\). The block is placed with \(A B\) perpendicular to the straight edge of a rough horizontal table. \(A X\) is in contact with the table and \(X B\) overhangs the edge (see diagram). The block is released from rest in this position, and it rotates without slipping about a horizontal axis through \(X\).

1. Find the moment of inertia of the block about the axis of rotation. For the instant when \(X G\) is horizontal,
2. show that the angular acceleration of the block is \(\frac { 3 \sqrt { 5 } g } { 25 a }\),
3. find the angular speed of the block,
4. show that the force exerted by the table on the block has magnitude \(\frac { 2 \sqrt { 70 } } { 25 } m g\).