6.05b Circular motion: v=r*omega and a=v^2/r

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A hollow container consists of a smooth circular cylinder of radius 0.5 m , and a smooth hollow cone of semi-vertical angle \(65 ^ { \circ }\) and radius 0.5 m . The container is fixed with its axis vertical and with the cone below the cylinder. A steel ball of weight 1 N moves with constant speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle inside the container. The ball is in contact with both the cylinder and the cone (see Fig. 1). Fig. 2 shows the forces acting on the ball, i.e. its weight and the forces of magnitudes \(R \mathrm {~N}\) and \(S \mathrm {~N}\) exerted by the container at the points of contact. Given that the radius of the ball is negligible compared with the radius of the cylinder, find \(R\) and \(S\).

3 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_637_572_264_788} One end of a light inextensible string is attached to a point \(C\). The other end is attached to a point \(D\), which is 1.1 m vertically below \(C\). A small smooth ring \(R\), of mass 0.2 kg , is threaded on the string and moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with centre at \(O\) and radius 1.2 m , where \(O\) is 0.5 m vertically below \(D\) (see diagram).

1. Show that the tension in the string is 1.69 N , correct to 3 significant figures.
2. Find the value of \(v\).

4
A particle of mass 0.12 kg is moving on the smooth inside surface of a fixed hollow sphere of radius 0.5 m . The particle moves in a horizontal circle whose centre is 0.3 m below the centre of the sphere (see diagram).

1. Show that the force exerted by the sphere on the particle has magnitude 2 N .
2. Find the speed of the particle.
3. Find the time taken for the particle to complete one revolution.

3 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-3_456_511_260_817} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T \mathrm {~N}\). The acceleration of the particle has magnitude \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(\tan \theta = 0.75\) and find the value of \(T\).
2. Find the speed of the particle.

5 One end of a light elastic string of natural length 0.3 m and modulus of elasticity 6 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg , which moves on the plane in a circular path with centre \(O\). The angular speed of \(P\) is \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. For the case \(\omega = 5\), calculate the extension of the string.
2. Express the extension of the string in terms of \(\omega\), and hence find the set of possible value of \(\omega\).

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small sphere \(S\) of mass \(m \mathrm {~kg}\) is moving inside a smooth hollow bowl whose axis is vertical and whose sloping side is inclined at \(60 ^ { \circ }\) to the horizontal. \(S\) moves with constant speed in a horizontal circle of radius 0.6 m (see Fig. 1). \(S\) is in contact with both the plane base and the sloping side of the bowl (see Fig. 2).

1. Given that the magnitudes of the forces exerted on \(S\) by the base and sloping side of the bowl are equal, calculate the speed of \(S\).
2. Given instead that \(S\) is on the point of losing contact with one of the surfaces, find the angular speed of \(S\).

7 Particles \(P\) and \(Q\), of masses 0.8 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string which passes through a small hole in a smooth horizontal table of negligible thickness. \(P\) moves with constant angular speed \(6.25 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a circular path on the surface of the table.

1. It is given that \(Q\) is stationary and that the part of string attached to \(Q\) is vertical. Calculate the radius of the path of \(P\), and find the speed of \(P\).
2. It is given instead that the part of string attached to \(Q\) is inclined at \(60 ^ { \circ }\) to the vertical, and that \(Q\) moves in a horizontal circular path below the table, also with constant angular speed \(6.25 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Calculate the total length of the string.
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4 \includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_170_616_1649_767} A small sphere \(S\) of mass \(m \mathrm {~kg}\) is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(S\) moves with constant speed in a horizontal circle of radius 0.4 m and is in contact with both the plane base and the curved surface of the cylinder (see diagram).

1. Given that the horizontal and vertical forces exerted on \(S\) by the cylinder have equal magnitudes, calculate the speed of \(S\). \(S\) is now attached to the centre of the base of the cylinder by a horizontal light elastic string of natural length 0.25 m and modulus of elasticity 13 N . The sphere \(S\) is set in motion and moves in a horizontal circle with constant angular speed \(\omega \mathrm { rads } ^ { - 1 }\) and is in contact with both the plane base and the curved surface of the cylinder.
2. It is given that the magnitudes of the horizontal and vertical forces exerted on \(S\) by the cylinder are equal if \(\omega = 8\). Calculate \(m\).
3. For the value of \(m\) found in part (ii), find the least possible value of \(\omega\) for the motion.

6 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_259_890_584_630} One end of a light inextensible string of length 0.2 m is attached to a fixed point \(A\) which is above a smooth horizontal table. A particle \(P\) of mass 0.3 kg is attached to the other end of the string. \(P\) moves on the table in a horizontal circle, with the string taut and making an angle of \(60 ^ { \circ }\) with the downward vertical (see diagram).

1. Calculate the tension in the string if the speed of \(P\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. For the motion as described, show that the angular speed of \(P\) cannot exceed \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and hence find the greatest possible value for the kinetic energy of \(P\).

1 A small sphere of mass 0.4 kg moves with constant speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle inside a smooth fixed hollow cylinder of diameter 0.6 m . The axis of the cylinder is vertical, and the sphere is in contact with both the horizontal base and the vertical curved surface of the cylinder.

1. Calculate the magnitude of the force exerted on the sphere by the vertical curved surface of the cylinder.
2. Hence show that the magnitude of the total force exerted on the sphere by the cylinder is 5 N .

6 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_291_993_1238_575} A uniform solid cone of height 0.6 m and mass 0.5 kg has its axis of symmetry vertical and its vertex \(V\) uppermost. The semi-vertical angle of the cone is \(60 ^ { \circ }\) and the surface is smooth. The cone is fixed to a horizontal surface. A particle \(P\) of mass 0.2 kg is connected to \(V\) by a light inextensible string of length 0.4 m (see diagram).

1. Calculate the height, above the horizontal surface, of the centre of mass of the cone with the particle. \(P\) is set in motion, and moves with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a circular path on the surface of the cone.
2. Show that the tension in the string is 1.96 N , and calculate the magnitude of the force exerted on \(P\) by the cone.
3. Find the speed of \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-2_631_531_1117_806} A smooth hollow cylinder of internal radius 0.3 m is fixed with its axis vertical. One end of a light inextensible string of length 0.5 m is fixed to a point \(A\) on the axis. The other end of the string is attached to a particle \(P\) of mass 0.2 kg which moves in a horizontal circle on the surface of the cylinder (see diagram).

1. Find the tension in the string.
2. Find the least angular speed of \(P\) for which the motion is possible.
3. Calculate the magnitude of the force exerted on \(P\) by the cylinder given that the speed of \(P\) is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-4_357_776_260_680} A small bead \(B\) of mass \(m \mathrm {~kg}\) moves with constant speed in a horizontal circle on a fixed smooth wire. The wire is in the form of a circle with centre \(O\) and radius 0.4 m . One end of a light elastic string of natural length 0.4 m and modulus of elasticity \(42 m \mathrm {~N}\) is attached to \(B\). The other end of the string is attached to a fixed point \(A\) which is 0.3 m vertically above \(O\) (see diagram).

1. Show that the vertical component of the contact force exerted by the wire on the bead is 3.7 mN upwards.
2. Given that the contact force has zero horizontal component, find the angular speed of \(B\).
3. Given instead that the horizontal component of the contact force has magnitude \(2 m \mathrm {~N}\), find the two possible speeds of \(B\). The string is now removed. \(B\) again moves on the wire in a horizontal circle with constant speed. It is given that the vertical and horizontal components of the contact force exerted by the wire on the bead have equal magnitudes.
4. Find the speed of \(B\). \end{document}

4 One end of a light inextensible string of length 2.4 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(0.2 \mathrm {~kg} . P\) moves with constant speed in a horizontal circle which has its centre vertically below \(A\), with the string taut and making an angle of \(60 ^ { \circ }\) with the vertical.

1. Find the speed of \(P\). The string of length 2.4 m is removed, and \(P\) is now connected to \(A\) by a light inextensible string of length 1.2 m . The particle \(P\) moves with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with its centre vertically below \(A\).
2. Calculate the angle between the string and the vertical.

1 A particle \(P\) of mass 0.6 kg is on the rough surface of a horizontal disc with centre \(O\). The distance \(O P\) is 0.4 m . The disc and \(P\) rotate with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis which passes through \(O\). Find the magnitude of the frictional force which the disc exerts on the particle, and state the direction of this force.

3 A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.5 m . The point \(A\) is 0.3 m above a smooth horizontal surface. The particle \(P\) moves in a horizontal circle on the surface with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Calculate the tension in the string. \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-05_67_1569_486_328}
2. Find the magnitude of the force exerted by the surface on \(P\).

1 A railway engine of mass 50000 kg travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal circular track of radius 1250 m . Find the magnitude of the horizontal force on the engine.

2 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-2_456_871_1228_635} An aircraft flies horizontally at a constant speed of \(220 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Initially it is flying due east. On reaching a point \(A\) it flies in a circular arc from \(A\) to \(B\), taking 50 s . At \(B\) the aircraft is flying due south (see diagram).

1. Show that the radius of the arc is approximately 7000 m .
2. Find the magnitude of the acceleration of the aircraft while it is flying between \(A\) and \(B\).

2 A horizontal turntable rotates with constant angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A particle of mass 0.06 kg is placed on the turntable at a point 0.25 m from its centre. The coefficient of friction between the particle and the turntable is \(\mu\). As the turntable rotates, the particle moves with the turntable and no sliding takes place.

1. Find the vertical and horizontal components of the contact force exerted on the particle by the turntable.
2. Show that \(\mu \geqslant 0.225\).

3 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-2_892_412_1217_865} A hollow cylinder of radius 0.35 m has a smooth inner surface. The cylinder is fixed with its axis vertical. One end of a light inextensible string of length 1.25 m is attached to a fixed point \(O\) on the axis of the cylinder. A particle \(P\) of mass 0.24 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle, in contact with the inner surface of the cylinder, and with the string taut (see diagram).

1. Find the tension in the string.
2. Given that the magnitude of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the force exerted on \(P\) by the cylinder.

5 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-4_495_1405_264_370} \(A B C D\) is a central cross-section of a uniform rectangular block of mass 35 kg . The lengths of \(A B\) and \(B C\) are 1.2 m and 0.8 m respectively. The block is held in equilibrium by a rope, one end of which is attached to the point \(E\) of a rough horizontal floor. The other end of the rope is attached to the block at \(A\). The rope is in the same vertical plane as \(A B C D\), and \(E A B\) is a straight line making an angle of \(20 ^ { \circ }\) with the horizontal (see diagram).

1. Show that the tension in the rope is 187 N , correct to the nearest whole number.
2. The block is on the point of slipping. Find the coefficient of friction between the block and the floor.

5 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_593_828_1530_660} A horizontal disc of radius 0.5 m is rotating with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed vertical axis through its centre \(O\). One end of a light inextensible string of length 0.8 m is attached to a point \(A\) of the circumference of the disc. A particle \(P\) of mass 0.4 kg is attached to the other end of the string. The string is taut and the system rotates so that the string is always in the same vertical plane as the radius \(O A\) of the disc. The string makes a constant angle \(\theta\) with the vertical (see diagram). The speed of \(P\) is 1.6 times the speed of \(A\).

1. Show that \(\sin \theta = \frac { 3 } { 8 }\).
2. Find the tension in the string.
3. Find the value of \(\omega\).

1 A horizontal circular disc rotates with constant angular speed \(9 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about its centre \(O\). A particle of mass 0.05 kg is placed on the disc at a distance 0.4 m from \(O\). The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc.

6 A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m . One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is \(T \mathrm {~N}\), and the bead rotates with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle about the vertical axis through \(A\) and \(C\).

1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m , calculate \(\omega\), and hence find the kinetic energy of \(B\).
2. Given instead that angle \(A B C = 90 ^ { \circ }\), and that \(A B\) makes an angle \(\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)\) with the vertical, calculate \(T\) and \(\omega\).

5 \includegraphics[max width=\textwidth, alt={}, center]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-3_927_1022_689_559} One end of a light inextensible string of length 1.2 m is attached to a fixed point \(O\) on a smooth horizontal surface. Particles \(P , Q\) and \(R\) are attached to the string so that \(O P = P Q = Q R = 0.4 \mathrm {~m}\). The particles rotate in horizontal circles about \(O\) with constant angular speed \(\omega \operatorname { rads } ^ { - 1 }\) and with \(O , P\), \(Q\) and \(R\) in a straight line (see diagram). \(R\) has mass 0.2 kg , and the tensions in the parts of the string attached to \(Q\) are 6 N and 10 N .

1. Show that \(\omega = 5\).
2. Calculate the mass of \(Q\).
3. Given that the kinetic energy of \(P\) is equal to the kinetic energy of \(R\), calculate the tension in the part of the string attached to \(O\).