6.05c Horizontal circles: conical pendulum, banked tracks

4 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-3_576_826_258_662} A hollow cone with semi-vertical angle \(45 ^ { \circ }\) is fixed with its axis vertical and its vertex \(O\) downwards. A particle \(P\) of mass 0.3 kg moves in a horizontal circle on the inner surface of the cone, which is smooth. \(P\) is attached to one end of a light inextensible string of length 1.2 m . The other end of the string is attached to the cone at \(O\) (see diagram). The string is taut and rotates at a constant angular speed of \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the acceleration of \(P\).
2. Find the tension in the string and the force exerted on \(P\) by the cone.

5
A toy aircraft of mass 0.5 kg is attached to one end of a light inextensible string of length 9 m . The other end of the string is attached to a fixed point \(O\). The aircraft moves with constant speed in a horizontal circle. The string is taut, and makes an angle of \(60 ^ { \circ }\) with the upward vertical at \(O\) (see diagram). In a simplified model of the motion, the aircraft is treated as a particle and the force of the air on the aircraft is taken to act vertically upwards with magnitude 8 N . Find

1. the tension in the string,
2. the speed of the aircraft.

7 One end of a light inextensible string of length 0.15 m is attached to a fixed point which is above a smooth horizontal surface. A particle of mass 0.5 kg is attached to the other end of the string. The particle moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with the string taut and making an angle of \(\theta ^ { \circ }\) with the downward vertical.

1. Given that \(\theta = 60\) and that the particle is not in contact with the surface, find \(v\).
2. Given instead that \(\theta = 45\) and \(v = 0.9\), and that the particle is in contact with the surface, find
   1. the tension in the string,
   2. the force exerted by the surface on the particle.

2 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-2_561_570_1274_790} A particle of mass 0.15 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 magnitude of the acceleration of the particle is \(7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The string makes an angle of \(\theta ^ { \circ }\) with the downward vertical, as shown in the diagram. Find

1. the value of \(\theta\) to the nearest whole number,
2. the tension in the string,
3. the speed of the particle.

3 \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-3_437_567_269_788} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) moves with constant speed in a horizontal circle, with the string taut and inclined at \(35 ^ { \circ }\) to the vertical. \(O P\) rotates with angular speed \(2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about the vertical axis through \(O\) (see diagram). Find

1. the value of \(L\),
2. the speed of \(P\) in \(\mathrm { m } \mathrm { s } ^ { - 1 }\).

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.

4 \(A B\) is the diameter of a uniform semicircular lamina which has radius 0.3 m and mass 0.4 kg . The lamina is hinged to a vertical wall at \(A\) with \(A B\) inclined at \(30 ^ { \circ }\) to the vertical. One end of a light inextensible string is attached to the lamina at \(B\) and the other end of the string is attached to the wall vertically above \(A\). The lamina is in equilibrium in a vertical plane perpendicular to the wall with the string horizontal (see diagram).

1. Show that the tension in the string is 2.00 N correct to 3 significant figures.
2. Find the magnitude and direction of the force exerted on the lamina by the hinge. \includegraphics[max width=\textwidth, alt={}, center]{5a2248f6-3ef9-4e69-90cf-4d6a2351be14-3_956_540_258_804} A small ball \(B\) of mass 0.4 kg is attached to fixed points \(P\) and \(Q\) on a vertical axis by two light inextensible strings of equal length. Both strings are taut and each is inclined at \(30 ^ { \circ }\) to the vertical. The ball moves in a horizontal circle (see diagram).

7 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-4_713_933_258_605} A narrow groove is cut along a diameter in the surface of a horizontal disc with centre \(O\). Particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, lie in the groove, and the coefficient of friction between each of the particles and the groove is \(\mu\). The particles are attached to opposite ends of a light inextensible string of length 1 m . The disc rotates with angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis passing through \(O\) and the particles move in horizontal circles (see diagram).

1. Given that \(\mu = 0.36\) and that both \(P\) and \(Q\) move in the same horizontal circle of radius 0.5 m , calculate the greatest possible value of \(\omega\) and the corresponding tension in the string.
2. Given instead that \(\mu = 0\) and that the tension in the string is 0.48 N , calculate
   1. the radius of the circle in which \(P\) moves and the radius of the circle in which \(Q\) moves,
   2. the speeds of the particles.

3 \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-3_385_1154_253_497} A particle \(P\) of mass 0.5 kg is attached to the vertex \(V\) of a fixed solid cone by a light inextensible string. \(P\) lies on the smooth curved surface of the cone and moves in a horizontal circle of radius 0.1 m with centre on the axis of the cone. The cone has semi-vertical angle \(60 ^ { \circ }\) (see diagram).

1. Calculate the speed of \(P\), given that the tension in the string and the contact force between the cone and \(P\) have the same magnitude.
2. Calculate the greatest angular speed at which \(P\) can move on the surface of the cone.

1 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-2_381_1079_255_534} A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string. The string is inclined at \(60 ^ { \circ }\) to the vertical. \(P\) moves with constant speed in a horizontal circle of radius 0.2 m . The centre of the circle is vertically below \(A\) (see diagram).

1. Show that the tension in the string is 8 N .
2. Calculate the speed of the particle.

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 \(S\).

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.

4 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of weight 6 N . Another light inextensible string of length 0.5 m connects \(P\) to a fixed point \(B\) which is 0.8 m vertically below \(A\). The particle \(P\) moves with constant speed in a horizontal circle with centre at the mid-point of \(A B\). Both strings are taut.

1. Calculate the speed of \(P\) when the tension in the string \(B P\) is 2 N .
2. Show that the angular speed of \(P\) must exceed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

7 A particle \(P\) of mass 0.7 kg is attached to one end of a light inextensible string of length 0.5 m . The other end of the string is attached to a fixed point \(A\) which is \(h \mathrm {~m}\) above a smooth horizontal surface. \(P\) moves in contact with the surface with uniform circular motion about the point on the surface which is vertically below \(A\).

1. Given that \(h = 0.14\), find an inequality for the angular speed of \(P\).
2. Given instead that the magnitude of the force exerted by the surface on \(P\) is 1.4 N and that the speed of \(P\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate the tension in the string and the value of \(h\).

6 \includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-4_479_499_255_824} \(O A\) is a rod which rotates in a horizontal circle about a vertical axis through \(O\). A particle \(P\) of mass 0.2 kg is attached to the mid-point of a light inextensible string. One end of the string is attached to the \(\operatorname { rod }\) at \(A\) and the other end of the string is attached to a point \(B\) on the axis. It is given that \(O A = O B\), angle \(O A P =\) angle \(O B P = 30 ^ { \circ }\), and \(P\) is 0.4 m from the axis. The rod and the particle rotate together about the axis with \(P\) in the plane \(O A B\) (see diagram).

1. Calculate the tensions in the two parts of the string when the speed of \(P\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angular speed of the rod is increased to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and it is given that the system now rotates with angle \(O A P =\) angle \(O B P = 60 ^ { \circ }\).
2. Show that the tension in the part \(A P\) of the string is zero. {www.cie.org.uk} after the live examination series. }

1 A particle \(P\) of mass 0.2 kg moves with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on a smooth surface. \(P\) is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to the point on the surface which is the centre of the circular motion of \(P\).

1. Find the radius of this circle.
2. Find the modulus of elasticity of the string.

2
The ends of two light inextensible strings of length 0.7 m are attached to a particle \(P\). The other ends of the strings are attached to two fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\). The particle \(P\) moves in a horizontal circle which has its centre at the mid-point of \(A B\). Both strings are inclined at \(60 ^ { \circ }\) to the vertical. The tension in the string attached to \(A\) is 6 N and the tension in the string attached to \(B\) is 4 N (see diagram).

1. Find the mass of \(P\).
2. Calculate the speed of \(P\).