Questions — OCR M4 (100 questions)

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OCR M4 2002 January Q1
1 A wheel rotating about a fixed axis is slowing down with constant angular deceleration. Initially the angular speed is \(24 \mathrm { rad } \mathrm { s } ^ { - 1 }\). In the first 5 seconds the wheel turns through 96 radians.
  1. Find the angular deceleration.
  2. Find the total angle the wheel turns through before coming to rest.
OCR M4 2002 January Q2
2 A uniform solid of revolution is formed by rotating the region bounded by the \(x\)-axis, the line \(x = 1\) and the curve \(y = x ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), about the \(x\)-axis. The units are metres, and the density of the solid is \(5400 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). Find the moment of inertia of this solid about the \(x\)-axis.
OCR M4 2002 January Q3
3 A uniform rectangular lamina \(A B C D\) of mass 0.6 kg has sides \(A B = 0.4 \mathrm {~m}\) and \(A D = 0.3 \mathrm {~m}\). The lamina is free to rotate about a fixed horizontal axis which passes through \(A\) and is perpendicular to the lamina.
  1. Find the moment of inertia of the lamina about the axis.
  2. Find the approximate period of small oscillations in a vertical plane.
OCR M4 2002 January Q4
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.
OCR M4 2002 January Q5
5 The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 5\) and the curve \(y = \mathrm { e } ^ { x }\) for \(0 \leqslant x \leqslant \ln 5\), is occupied by a uniform lamina.
  1. Show that the centre of mass of this lamina has \(x\)-coordinate $$\frac { 5 } { 4 } \ln 5 - 1$$
  2. Find the \(y\)-coordinate of the centre of mass.
OCR M4 2002 January Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{98647526-b52a-4316-9a09-48d756b8f599-3_117_913_251_630} An arm on a fairground ride is modelled as a uniform rod \(A B\), of mass 75 kg and length 7.2 m , with a particle of mass 124 kg attached at \(B\). The arm can rotate about a fixed horizontal axis perpendicular to the rod and passing through the point \(P\) on the rod, where \(A P = 1.2 \mathrm {~m}\).
  1. Show that the moment of inertia of the arm about the axis is \(5220 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
  2. The arm is released from rest with \(A B\) horizontal, and a frictional couple of constant moment 850 N m opposes the motion. Find the angular speed of the arm when \(B\) is first vertically below \(P\).
OCR M4 2002 January Q7
7 At midnight, ship \(A\) is 70 km due north of ship \(B\). Ship \(A\) travels with constant velocity \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(140 ^ { \circ }\). Ship \(B\) travels with constant velocity \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(025 ^ { \circ }\).
  1. Find the magnitude and direction of the velocity of \(A\) relative to \(B\).
  2. Find the distance between the ships when they are at their closest, and find the time when this occurs.
OCR M4 2002 January Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{98647526-b52a-4316-9a09-48d756b8f599-3_493_748_1393_708} The diagram shows a uniform rod \(A B\), of mass \(m\) and length \(2 a\), free to rotate in a vertical plane about a fixed horizontal axis through \(A\). A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\). The string joins \(B\) to a light ring \(R\) which slides along a smooth horizontal wire fixed at a height \(a\) above \(A\) and in the same vertical plane as \(A B\). The string \(B R\) remains vertical. The angle between \(A B\) and the horizontal is denoted by \(\theta\), where \(0 < \theta < \pi\).
  1. Taking the reference level for gravitational potential energy to be the horizontal through \(A\), show that the total potential energy of the system is $$m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) .$$
  2. Find the three values of \(\theta\) for which the system is in equilibrium.
  3. For each position of equilibrium, determine whether it is stable or unstable.
OCR M4 2004 January Q1
1 A wheel is rotating about a fixed axis, and is slowing down with constant angular deceleration \(0.3 \mathrm { rad } \mathrm { s } ^ { - 2 }\).
  1. Find the angle the wheel turns through as its angular speed changes from \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  2. Find the time taken for the wheel to make its final complete revolution before coming to rest.
OCR M4 2004 January Q2
2 A rod \(A B\) of variable density has length 2 m . At a distance \(x\) metres from \(A\), the rod has mass per unit length ( \(0.7 - 0.3 x ) \mathrm { kg } \mathrm { m } ^ { - 1 }\). Find the distance of the centre of mass of the rod from \(A\).
OCR M4 2004 January Q3
3 From a speedboat, a ship is sighted on a bearing of \(045 ^ { \circ }\). The ship has constant velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(120 ^ { \circ }\). The speedboat travels in a straight line with constant speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and intercepts the ship.
  1. Find the bearing of the course of the speedboat.
  2. Find the magnitude of the velocity of the ship relative to the speedboat.
OCR M4 2004 January Q4
4 The region between the curve \(y = \frac { x ^ { 2 } } { a }\) and the \(x\)-axis for \(0 \leqslant x \leqslant a\) is occupied by a uniform lamina with mass \(m\). Show that the moment of inertia of this lamina about the \(x\)-axis is \(\frac { 1 } { 7 } m a ^ { 2 }\).
OCR M4 2004 January Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-2_618_627_1594_743} A uniform circular disc has mass 4 kg , radius 0.6 m and centre \(C\). The disc can rotate in a vertical plane about a fixed horizontal axis which is perpendicular to the disc and which passes through the point \(A\) on the disc, where \(A C = 0.4 \mathrm {~m}\). A frictional couple of constant moment 4.8 Nm opposes the motion. The disc is released from rest with \(A C\) horizontal (see diagram).
  1. Find the moment of inertia of the disc about the axis through \(A\).
  2. Find the angular acceleration of the disc immediately after it is released.
  3. Find the angular speed of the disc when \(C\) is first vertically below \(A\).
OCR M4 2004 January Q6
6 A rigid body consists of a uniform rod \(A B\), of mass 15 kg and length 2.8 m , with a particle of mass 5 kg attached at \(B\). The body rotates without resistance in a vertical plane about a fixed horizontal axis through \(A\).
  1. Find the distance of the centre of mass of the body from \(A\).
  2. Find the moment of inertia of the body about the axis.
    \includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-3_475_682_680_719} At one instant, \(A B\) makes an acute angle \(\theta\) with the downward vertical, the angular speed of the body is \(1.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the angular acceleration of the body is \(3.5 \mathrm { rad } \mathrm { s } ^ { - 2 }\) (see diagram).
  3. Show that \(\sin \theta = 0.8\).
  4. Find the components, parallel and perpendicular to \(B A\), of the force acting on the body at \(A\).
    [0pt] [Question 7 is printed overleaf.]
    \includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-4_949_1112_281_550} A small bead \(B\), of mass \(m\), slides on a smooth circular hoop of radius \(a\) and centre \(O\) which is fixed in a vertical plane. A light elastic string has natural length \(2 a\) and modulus of elasticity \(m g\); one end is attached to \(B\), and the other end is attached to a light ring \(R\) which slides along a smooth horizontal wire. The wire is in the same vertical plane as the hoop, and at a distance \(2 a\) above \(O\). The elastic string \(B R\) is always vertical, and \(O B\) makes an angle \(\theta\) with the downward vertical (see diagram).
  5. Show that \(\theta = 0\) is a position of stable equilibrium.
  6. Find the approximate period of small oscillations about the equilibrium position \(\theta = 0\).
OCR M4 2003 June Q1
1 A propeller shaft has constant angular acceleration. It turns through 160 radians as its angular speed increases from \(15 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find
  1. the angular acceleration of the propeller shaft,
  2. the time taken for this increase in angular speed.
OCR M4 2003 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-2_462_490_610_834} The diagram shows a uniform lamina \(A B C D E F\) in which all the corners are right angles. The mass of the lamina is \(3 m\).
  1. Show that the moment of inertia of the lamina about \(A B\) is \(3 m a ^ { 2 }\).
  2. Find the moment of inertia of the lamina about an axis perpendicular to the lamina and passing through \(A\).
OCR M4 2003 June Q3
3 A uniform rod, of mass 0.75 kg and length 1.6 m , rotates in a vertical plane about a fixed horizontal axis through one end. A frictional couple of constant moment opposes the motion. The rod is released from rest in a horizontal position and, when the rod is first vertical, its angular speed is \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Find the magnitude of the frictional couple.
    \includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-2_584_527_1798_822} A disc is rotating about the same axis. The moment of inertia of the disc about the axis is \(0.56 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). When the rod is vertical, the disc has angular speed \(4.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in the opposite direction to that of the rod (see diagram). At this instant the rod hits a magnetic catch \(C\) on the disc and becomes attached to the disc.
  2. Find the angular speed of the rod and disc immediately after they have become attached.
OCR M4 2003 June Q4
4 A cruise ship \(C\) is sailing due north at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat \(B\), initially 2000 m due west of \(C\), sails with constant speed \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight line course which takes it as close as possible to \(C\).
  1. Find the bearing of the direction in which \(B\) sails.
  2. Find the shortest distance between \(B\) and \(C\) in the subsequent motion.
OCR M4 2003 June Q5
5 The region bounded by the \(x\)-axis, the line \(x = 8\) and the curve \(y = x ^ { \frac { 1 } { 3 } }\) for \(0 \leqslant x \leqslant 8\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. The unit of length is the metre, and the density of the solid is \(350 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).
  1. Show that the mass of the solid is \(6720 \pi \mathrm {~kg}\).
  2. Find the \(x\)-coordinate of the centre of mass of the solid.
  3. Find the moment of inertia of the solid about the \(x\)-axis.
OCR M4 2003 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-3_468_550_1201_824} A wheel consists of a uniform circular disc, with centre \(O\), mass 0.08 kg and radius 0.35 m , with a particle \(P\) of mass 0.24 kg attached to a point on the circumference. The wheel is rotating without resistance in a vertical plane about a fixed horizontal axis through \(O\) (see diagram).
  1. Find the moment of inertia of the wheel about the axis.
  2. Find the distance of the centre of mass of the wheel from the axis. At an instant when \(O P\) is horizontal and the angular speed of the wheel is \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find
  3. the angular acceleration of the wheel,
  4. the magnitude of the force acting on the wheel at \(O\).
OCR M4 2003 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-4_557_1036_278_553} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is pivoted to a fixed point at \(A\) and is free to rotate in a vertical plane. Two fixed vertical wires in this plane are a distance \(6 a\) apart and the point \(A\) is half-way between the two wires. Light smooth rings \(R _ { 1 }\) and \(R _ { 2 }\) slide on the wires and are connected to \(B\) by light elastic strings, each of natural length \(a\) and modulus of elasticity \(\frac { 1 } { 4 } m g\). The strings \(B R _ { 1 }\) and \(B R _ { 2 }\) are always horizontal and the angle between \(A B\) and the upward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram).
  1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a \left( 1 + \cos \theta + \sin ^ { 2 } \theta \right) .$$
  2. Given that \(\theta = 0\) is a position of stable equilibrium, find the approximate period of small oscillations about this position.
OCR M4 2004 June Q1
1 Two flywheels \(P\) and \(Q\) are rotating, in opposite directions, about the same fixed axis. The angular speed of \(P\) is \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the angular speed of \(Q\) is \(30 \mathrm { rad } \mathrm { s } ^ { - 1 }\). The flywheels lock together, and after this they both rotate with angular speed \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in the direction in which \(P\) was originally rotating. The moment of inertia of \(P\) about the axis is \(0.64 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). Find the moment of inertia of \(Q\) about the axis.
OCR M4 2004 June Q2
2 A uniform rectangular lamina has mass \(m\) and sides of length \(3 a\) and \(4 a\), and rotates freely about a fixed horizontal axis. The axis is perpendicular to the lamina and passes through a corner. The lamina makes small oscillations in its own plane, as a compound pendulum.
  1. Find the moment of inertia of the lamina about the axis.
  2. Find the approximate period of the small oscillations.
OCR M4 2004 June Q3
3 The region between the curve \(y = x \sqrt { } ( 3 - x )\) and the \(x\)-axis for \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.
OCR M4 2004 June Q4
4 A uniform solid sphere, of mass 14 kg and radius 0.25 m , is rotating about a fixed axis which is a diameter of the sphere. A couple of constant moment 4.2 Nm about the axis, acting in the direction of rotation, is applied to the sphere.
  1. Find the angular acceleration of the sphere. During a time interval of 30 seconds the sphere rotates through 7500 radians.
  2. Find the angular speed of the sphere at the start of the time interval.
  3. Find the angular speed of the sphere at the end of the time interval.
  4. Find the work done by the couple during the time interval.