6.05a Angular velocity: definitions

4

1. The region \(T\) is bounded by the \(x\)-axis, the line \(y = k x\) for \(a \leqslant x \leqslant 3 a\), the line \(x = a\) and the line \(x = 3 a\), where \(k\) and \(a\) are positive constants. A uniform frustum of a cone is formed by rotating \(T\) about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this frustum.
2. A uniform lamina occupies the region (shown in Fig. 4) bounded by the \(x\)-axis, the curve \(y = 16 \left( 1 - x ^ { - \frac { 1 } { 3 } } \right)\) for \(1 \leqslant x \leqslant 8\) and the line \(x = 8\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86d79489-aec1-4c94-bef6-45b007f818a0-4_368_519_1439_772} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Find the coordinates of the centre of mass of this lamina. A hole is made in the lamina by cutting out a circular disc of area 5 square units. This causes the centre of mass of the lamina to move to the point \(( 5,3 )\).
   2. Find the coordinates of the centre of the hole.

2

1. A fixed solid sphere with a smooth surface has centre O and radius 0.8 m . A particle P is given a horizontal velocity of \(1.2 \mathrm {~ms} ^ { - 1 }\) at the highest point on the sphere, and it moves on the surface of the sphere in part of a vertical circle of radius 0.8 m .
   1. Find the radial and tangential components of the acceleration of P at the instant when OP makes an angle \(\frac { 1 } { 6 } \pi\) radians with the upward vertical. (You may assume that P is still in contact with the sphere.)
   2. Find the speed of P at the instant when it leaves the surface of the sphere.
2. Two fixed points R and S are 2.5 m apart with S vertically below R . A particle Q of mass 0.9 kg is connected to R and to S by two light inextensible strings; Q is moving in a horizontal circle at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with both strings taut. The radius of the circle is 2.4 m and the centre C of the circle is 0.7 m vertically below S, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3f674569-7e99-4ba8-84f1-a1eb438e30ed-2_547_720_1946_644} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Find the tension in the string RQ and the tension in the string \(S Q\).

2

1. A particle P of mass 0.6 kg is connected to a fixed point by a light inextensible string of length 2.8 m . The particle P moves in a horizontal circle as a conical pendulum, with the string making a constant angle of \(55 ^ { \circ }\) with the vertical.
   1. Find the tension in the string.
   2. Find the speed of P .
2. A turntable has a rough horizontal surface, and it can rotate about a vertical axis through its centre O . While the turntable is stationary, a small object Q of mass 0.5 kg is placed on the turntable at a distance of 1.4 m from O . The turntable then begins to rotate, with a constant angular acceleration of \(1.12 \mathrm { rad } \mathrm { s } ^ { - 2 }\). Let \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) be the angular speed of the turntable. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-3_517_522_870_769} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Given that Q does not slip, find the components \(F _ { 1 }\) and \(F _ { 2 }\) of the frictional force acting on Q perpendicular and parallel to QO (see Fig. 2). Give your answers in terms of \(\omega\) where appropriate. The coefficient of friction between Q and the turntable is 0.65 .
   2. Find the value of \(\omega\) when Q is about to slip.
   3. Find the angle which the frictional force makes with QO when Q is about to slip.

2 A fixed hollow sphere with centre O has an inside radius of 2.7 m . A particle P of mass 0.4 kg moves on the smooth inside surface of the sphere. At first, P is moving in a horizontal circle with constant speed, and OP makes a constant angle of \(60 ^ { \circ }\) with the vertical (see Fig. 2.1). \begin{figure}[h] \end{figure}

1. Find the normal reaction acting on P .
2. Find the speed of P . The particle P is now placed at the lowest point of the sphere and is given an initial horizontal speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves in part of a vertical circle. When OP makes an angle \(\theta\) with the upward vertical and P is still in contact with the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\) (see Fig. 2.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-3_716_778_1653_696} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. Find \(v ^ { 2 }\) in terms of \(\theta\).
4. Show that \(R = 4.16 - 11.76 \cos \theta\).
5. Find the speed of P at the instant when it leaves the surface of the sphere.

2 A particle P of mass 0.3 kg is connected to a fixed point O by a light inextensible string of length 4.2 m . Firstly, P is moving in a horizontal circle as a conical pendulum, with the string making a constant angle with the vertical. The tension in the string is 3.92 N .

1. Find the angle which the string makes with the vertical.
2. Find the speed of P . P now moves in part of a vertical circle with centre O and radius 4.2 m . When the string makes an angle \(\theta\) with the downward vertical, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2). You are given that \(v = 8.4\) when \(\theta = 60 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2a4afead-e772-4d86-bc8d-86ffa5bca507-2_382_648_1985_751} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
3. Find the tension in the string when \(\theta = 60 ^ { \circ }\).
4. Show that \(v ^ { 2 } = 29.4 + 82.32 \cos \theta\).
5. Find \(\theta\) at the instant when the string becomes slack.

1 A fixed solid sphere has centre O and radius 2.6 m . A particle P of mass 0.65 kg moves on the smooth surface of the sphere. The particle P is set in motion with horizontal velocity \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the highest point of the sphere, and moves in part of a vertical circle. When OP makes an angle \(\theta\) with the upward vertical, and P is still in contact with the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 52.92 - 50.96 \cos \theta\).
2. Find, in terms of \(\theta\), the normal reaction acting on P .
3. Find the speed of P at the instant when it leaves the surface of the sphere. The particle P is now attached to one end of a light inextensible string, and the other end of the string is fixed to a point A , vertically above O , such that AP is tangential to the sphere, as shown in Fig. 1. P moves with constant speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with radius 2.4 m on the surface of the sphere. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3ec81c4e-e0fa-43d9-9c79-ef9df746be8f-2_1100_634_1089_753} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
4. Find the tension in the string and the normal reaction acting on P .

2 A hollow hemisphere has internal radius 2.5 m and is fixed with its rim horizontal and uppermost. The centre of the hemisphere is O . A small ball B of mass 0.4 kg moves in contact with the smooth inside surface of the hemisphere. At first, B is moving at constant speed in a horizontal circle with radius 1.5 m , as shown in Fig. 2.1. \begin{figure}[h] \end{figure}

1. Find the normal reaction of the hemisphere on \(B\).
2. Find the speed of \(\mathbf { B }\). The ball B is now released from rest on the inside surface at a point on the same horizontal level as O . It then moves in part of a vertical circle with centre O and radius 2.5 m , as shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-3_378_663_1427_740} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. Show that, when \(B\) is at its lowest point, the normal reaction is three times the weight of \(B\). For an instant when the normal reaction is twice the weight of \(\mathbf { B }\), find
4. the speed of \(\mathbf { B }\),
5. the tangential component of the acceleration of \(\mathbf { B }\).

2

1. A particle P of mass 0.2 kg is connected to a fixed point O by a light inextensible string of length 3.2 m , and is moving in a vertical circle with centre O and radius 3.2 m . Air resistance may be neglected. When P is at the highest point of the circle, the tension in the string is 0.6 N .
   1. Find the speed of P when it is at the highest point.
   2. For an instant when OP makes an angle of \(60 ^ { \circ }\) with the downward vertical, find
      (A) the radial and tangential components of the acceleration of P ,
      (B) the tension in the string.
2. A solid cone is fixed with its axis of symmetry vertical and its vertex V uppermost. The semivertical angle of the cone is \(36 ^ { \circ }\), and its surface is smooth. A particle Q of mass 0.2 kg is connected to V by a light inextensible string, and Q moves in a horizontal circle at constant speed, in contact with the surface of the cone, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-3_455_609_950_808} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The particle Q makes one complete revolution in 1.8 s , and the normal reaction of the cone on Q has magnitude 0.75 N .
   1. Find the tension in the string.
   2. Find the length of the string.

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.

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.

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.

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.

1 A wheel is rotating freely with angular speed \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed axis through its centre. The moment of inertia of the wheel about the axis is \(0.65 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). A couple of constant moment is applied to the wheel, and in the next 5 seconds the wheel rotates through 180 radians.

1. Find the angular acceleration of the wheel.
2. Find the moment of the couple about the axis.

1 The driveshaft of an electric motor begins to rotate from rest and has constant angular acceleration. In the first 8 seconds it turns through 56 radians.

1. Find the angular acceleration.
2. Find the angle through which the driveshaft turns while its angular speed increases from \(20 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(36 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

2 A rotating turntable is slowing down with constant angular deceleration. It makes 16 revolutions as its angular speed decreases from \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to rest.

1. Find the angular deceleration of the turntable.
2. Find the angular speed of the turntable at the start of its last complete revolution before coming to rest.
3. Find the time taken for the turntable to make its last complete revolution before coming to rest.

5 The region bounded by the curve \(y = \sqrt { a x }\) for \(a \leqslant x \leqslant 4 a\) (where \(a\) is a positive constant), the \(x\)-axis, and the lines \(x = a\) and \(x = 4 a\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution of mass \(m\).

1. Show that the moment of inertia of this solid about the \(x\)-axis is \(\frac { 7 } { 5 } m a ^ { 2 }\). The solid is free to rotate about a fixed horizontal axis along the line \(y = a\), and makes small oscillations as a compound pendulum.
2. Find, in terms of \(a\) and \(g\), the approximate period of these small oscillations. \includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-3_734_862_813_644} A uniform rectangular lamina \(A B C D\) has mass \(m\) and sides \(A B = 2 a\) and \(B C = 3 a\). The mid-point of \(A B\) is \(P\) and the mid-point of \(C D\) is \(Q\). The lamina is rotating freely in a vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the point \(X\) on \(P Q\) where \(P X = a\). Air resistance may be neglected. When \(Q\) is vertically above \(X\), the angular speed is \(\sqrt { \frac { 9 g } { 10 a } }\). When \(X Q\) makes an angle \(\theta\) with the upward vertical, the angular speed is \(\omega\), and the force acting on the lamina at \(X\) has components \(R\) parallel to \(P Q\) and \(S\) parallel to \(B A\) (see diagram).

1 A top is set spinning with initial angular speed \(83 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and it slows down with constant angular deceleration. When it has turned through 1000 radians, its angular speed is \(67 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the angular deceleration of the top.
2. Find the time taken, from the start, for the top to turn through 400 radians.

3 \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-2_664_623_904_760} A uniform circular disc has mass \(4 m\), radius \(2 a\) and centre \(O\). The points \(A\) and \(B\) are at opposite ends of a diameter of the disc, and the mid-point of \(O A\) is \(P\). A particle of mass \(m\) is attached to the disc at \(B\). The resulting compound pendulum is in a vertical plane and is free to rotate about a fixed horizontal axis passing through \(P\) and perpendicular to the disc (see diagram). The pendulum makes small oscillations.

1. Find the moment of inertia of the pendulum about the axis.
2. Find the approximate period of the small oscillations.

1 A wheel is rotating and is slowing down with constant angular deceleration. The initial angular speed is \(80 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and after 15 s the wheel has turned through 1020 radians.

1. Find the angular deceleration of the wheel.
2. Find the angle through which the wheel turns in the last 5 s before it comes to rest.
3. Find the total number of revolutions made by the wheel from the start until it comes to rest.

7 \includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-4_524_732_258_705} The diagram shows a uniform rectangular lamina \(A B C D\) with \(A B = 6 a , A D = 8 a\) and centre \(G\). The mass of the lamina is \(m\). The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through \(A\) and perpendicular to the lamina.

1. Find the moment of inertia of the lamina about this axis. The lamina is released from rest with \(A D\) horizontal and \(B C\) below \(A D\).
2. For an instant during the subsequent motion when \(A D\) is vertical, show that the angular speed of the lamina is \(\sqrt { \frac { 3 g } { 50 a } }\) and find its angular acceleration. At an instant when \(A D\) is vertical, the force acting on the lamina at \(A\) has magnitude \(F\).
3. By finding components parallel and perpendicular to \(G A\), or otherwise, show that \(F = \frac { \sqrt { 493 } } { 20 } \mathrm { mg }\).
   [0pt] [8]

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.

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
   1. immediately after the couple is first applied,
   2. when \(X Y\) is horizontal.

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.

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 \(\frac { 2 \sqrt { 70 } } { 25 } m g\).

1 A uniform square lamina, of mass 4.5 kg and side 0.6 m , is rotating about a fixed vertical axis which is perpendicular to the lamina and passes through its centre. A stationary particle becomes attached to the lamina at one of its corners, and this causes the angular speed of the lamina to change instantaneously from \(2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the mass of the particle. The lamina then slows down with constant angular deceleration. It turns through 36 radians as its angular speed reduces from \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to zero.
2. Find the time taken for the lamina to come to rest.