Vertical circle: speed at specific point

5 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The string is held taut with \(O P\) horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { \frac { 1 } { 3 } \mathrm { ag } }\) and starts to move in a vertical circle. \(P\) passes through the lowest point of the circle and reaches the point \(Q\) where \(O Q\) makes an angle \(\theta\) with the downward vertical. At \(Q\) the speed of \(P\) is \(\sqrt { \mathrm { kag } }\) and the tension in the string is \(\frac { 11 } { 6 } \mathrm { mg }\).

1. Find the value of \(k\) and the value of \(\cos \theta\).
   At \(Q\) the particle \(P\) becomes detached from the string.
2. In the subsequent motion, find the greatest height reached by \(P\) above the level of the lowest point of the circle.

1
\includegraphics[max width=\textwidth, alt={}, center]{5960a9cf-2c51-4c07-9973-c29604762df7-2_540_269_395_897} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length 3.2 m . The other end of the string is attached to a fixed point \(O\). The particle is held at rest, with the string taut and making an angle of \(15 ^ { \circ }\) with the vertical. It is then projected with velocity \(1.2 \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(O P\) and with a downwards component so that it begins to move in a vertical circle (see diagram). In the ensuing motion the string remains taut and the angle it makes with the downwards vertical through \(O\) is denoted by \(\theta ^ { \circ }\).

1. Find the speed of \(P\) at the point on its path vertically below \(O\).
2. Find the value of \(\theta\) at the point where \(P\) first comes to instantaneous rest.

1 One end of a light inextensible string of length 0.8 m is attached to a particle \(P\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a fixed point \(O\). Initially \(P\) hangs in equilibrium vertically below \(O\). It is then projected horizontally with a speed of \(5.3 \mathrm {~ms} ^ { - 1 }\) so that it moves in a vertical circular path with centre \(O\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{894be707-4f7b-4647-b209-805522556196-2_686_586_450_248} At a certain instant, \(P\) first reaches the point where the string makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downward vertical through \(O\).

1. Show that at this instant the speed of \(P\) is \(4.5 \mathrm {~ms} ^ { - 1 }\).
2. Find the magnitude and direction of the radial acceleration of \(P\) at this instant.
3. Find the magnitude of the tangential acceleration of \(P\) at this instant.

7 A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A particle, of mass \(m\), is attached to the other end. The particle is moving in a vertical circle, centre \(O\). When the particle is at \(B\), vertically above \(O\), the string is taut and the particle is moving with speed \(3 \sqrt { a g }\).
\includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-5_422_399_497_778}

1. Find, in terms of \(g\) and \(a\), the speed of the particle at the lowest point, \(A\), of its path.
2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(A\).

7 A small ball, of mass 3 kg , is suspended from a fixed point \(O\) by a light inextensible string of length 1.2 m . Initially, the string is taut and the ball is at the point \(P\), vertically below \(O\). The ball is then set into motion with an initial horizontal velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball moves in a vertical circle, centre \(O\). The point \(A\), on the circle, is such that angle \(A O P\) is \(25 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-5_663_702_660_701}

1. Find the speed of the ball at the point \(A\).
2. Find the tension in the string when the ball is at the point \(A\).

4 A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(P\) vertically below \(O\). The particle is then set into motion with a horizontal velocity \(U\) so that it moves in a complete vertical circle with centre \(O\). The point \(Q\) on the circle is such that \(\angle P O Q = 60 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-3_566_540_1797_751}

1. Find, in terms of \(g , l\) and \(U\), the speed of the particle at \(Q\).
2. Find, in terms of \(g , l , m\) and \(U\), the tension in the string when the particle is at \(Q\).
3. Find, in terms of \(g , l , m\) and \(U\), the tension in the string when the particle returns to \(P\).
   (2 marks)

8 A particle is attached to one end of a light inextensible string of length 3 metres. The other end of the string is attached to a fixed point \(O\). The particle is set into motion horizontally at point \(P\) with speed \(v\), so that it describes part of a vertical circle whose centre is \(O\). The point \(P\) is vertically below \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-16_510_334_493_861} The particle first comes momentarily to rest at the point \(Q\), where \(O Q\) makes an angle of \(15 ^ { \circ }\) to the vertical.

1. Find the value of \(v\).
2. When the particle is at rest at the point \(Q\), the tension in the string is 22 newtons. Find the mass of the particle.
   
   \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-17_2484_1709_223_153}

5 A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A particle, of mass \(m\), is attached to the other end of the string. The particle is moving in a vertical circle with centre \(O\). The point \(Q\) is the highest point of the particle's path. When the particle is at \(P\), vertically below \(O\), the string is taut and the particle is moving with speed \(7 \sqrt { a g }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-10_887_812_525_628}

1. Find, in terms of \(g\) and \(a\), the speed of the particle at the point \(Q\).
2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(Q\).

3
\includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-2_473_298_1037_884} A particle \(P\) of mass 1.5 kg is attached to one end of a light inextensible string of length 2.4 m . The other end of the string is attached to a fixed point \(O\). The particle is initially at rest directly below \(O\). A horizontal impulse of magnitude 9.3 Ns is applied to \(P\). In the subsequent motion the string remains taut and makes an angle of \(\theta\) radians with the downwards vertical at \(O\), as shown in the diagram.

1. Find the speed of \(P\) when \(\theta = \frac { 1 } { 6 } \pi\).
2. Determine whether \(P\) will reach the same horizontal level as \(O\).

2
\includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-02_810_743_831_644} A smooth wire is shaped into a circle of centre \(O\) and radius 0.8 m . The wire is fixed in a vertical plane. A small bead \(P\) of mass 0.03 kg is threaded on the wire and is projected along the wire from the highest point with a speed of \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(O P\) makes an angle \(\theta\) with the upward vertical the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).

1. Show that \(v ^ { 2 } = 33.32 - 15.68 \cos \theta\).
2. Prove that the bead is never at rest.
3. Find the maximum value of \(v\).
4. Write down the dimension of density. The workings of an oil pump consist of a right, solid cylinder which is partially submerged in oil. The cylinder is free to oscillate along its central axis which is vertical. If the base area of the pump is \(0.4 \mathrm {~m} ^ { 2 }\) and the density of the oil is \(920 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) then the period of oscillation of the pump is 0.7 s .
   A student assumes that the period of oscillation of the pump is dependent only on the density of the oil, \(\rho\), the acceleration due to gravity, \(g\), and the surface area, \(A\), of the circular base of the pump. The student attempts to test this assumption by stating that the period of oscillation, \(T\), is given by \(T = C \rho ^ { \alpha } g ^ { \beta } A ^ { \gamma }\) where \(C\) is a dimensionless constant.
5. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\).
6. Hence give the value of \(C\) to 3 significant figures.
7. Comment, with justification, on the assumption made by the student that the formula for the period of oscillation of the pump was dependent on only \(\rho , g\) and \(A\). A car of mass 1250 kg experiences a resistance to its motion of magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). At a point \(A\) on the road the car's speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it has an acceleration of magnitude \(0.54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At a point \(B\) on the road the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it has an acceleration of magnitude \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
8. Find the values of \(k\) and \(P\). The power is increased to 15 kW .
9. Calculate the maximum steady speed of the car on a straight horizontal road.