Vertical circle: string becomes slack

2 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string. The particle is held at the point \(A\) with the string taut. The angle between \(O A\) and the downward vertical is equal to \(\alpha\), where \(\cos \alpha = \frac { 4 } { 5 }\). The particle is projected from \(A\), perpendicular to the string in an upwards direction, with a speed \(\sqrt { 3 \text { ga } }\). It then moves along a circular path in a vertical plane. The string first goes slack when it makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\).

3 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 particle \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(A O B\) is \(90 ^ { \circ }\) and the speed of \(P\) is \(\sqrt { \frac { 4 } { 5 } \mathrm { ag } }\).

1. Find the value of \(\sin \theta\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\).
   \includegraphics[max width=\textwidth, alt={}, center]{3b50dc98-781e-4399-8165-ad5e3065df4b-06_846_767_258_689} An object is formed from a solid hemisphere, of radius \(2 a\), and a solid cylinder, of radius \(a\) and height \(d\). The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line \(O C\) forms a diameter of the base of the cylinder, where \(C\) is the centre of the plane face of the hemisphere and \(O\) is common to both circumferences (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(O C\) as shown, the centre of mass of the object is ( \(\mathrm { x } , \mathrm { y }\) ).

1 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 particle \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\alpha\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(\sqrt { 3 a g }\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical. Given that \(\cos \alpha = \frac { 4 } { 5 }\), find the value of \(\cos \theta\).

3 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 \(O\). When the particle is hanging vertically below \(O\), it is projected horizontally with speed \(u\) so that it begins to move along a circular path. When \(P\) is at the lowest point of its motion, the tension in the string is \(T\). When \(O P\) makes an angle \(\theta\) with the upward vertical, the tension in the string is \(S\).

1. Show that \(S = T - 3 m g ( 1 + \cos \theta )\).
   \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-04_2718_38_141_2010}
2. Given that \(u = \sqrt { 4 a g }\), find the value of \(\cos \theta\) when the string goes slack.
   \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-06_348_707_285_680} A light spring of natural length \(a\) and modulus of elasticity \(k m g\) is attached to a fixed point \(O\) on a smooth plane inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = \frac { 3 } { 4 }\). A particle of mass \(m\) is attached to the lower end of the spring and is held at the point \(A\) on the plane, where \(O A = 2 a\) and \(O A\) is along a line of greatest slope of the plane (see diagram). The particle is released from rest and is moving with speed \(V\) when it passes through the point \(B\) on the plane, where \(O B = \frac { 3 } { 2 } a\). The speed of the particle is \(\frac { 1 } { 2 } V\) when it passes through the point \(C\) on the plane, where \(O C = \frac { 3 } { 4 } a\). Find the value of \(k\).
   \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-06_2718_35_141_2012}

1 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 \(O P\) making an angle \(\alpha\) with the downward vertical, where \(\cos \alpha = \frac { 2 } { 3 }\). The particle \(P\) is projected perpendicular to \(O P\) in an upwards direction with speed \(\sqrt { 3 a g }\). It then starts to move along a circular path in a vertical plane. Find the cosine of the angle between the string and the upward vertical when the string first becomes slack.
\includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-03_671_455_255_845} A uniform lamina is in the form of a triangle \(A B C\) in which angle \(B\) is a right angle, \(\mathrm { AB } = 9 \mathrm { a }\) and \(\mathrm { BC } = 6 \mathrm { a }\). The point \(D\) is on \(B C\) such that \(\mathrm { BD } = \mathrm { x }\) (see diagram). The region \(A B D\) is removed from the lamina. The resulting shape \(A D C\) is placed with the edge \(D C\) on a horizontal surface and the plane \(A D C\) is vertical. Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium.

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A = l\) and \(O A\) is horizontal. The particle is then projected vertically downwards from \(A\) with speed \(\sqrt { 2 g l }\), as shown in Figure 4 . When the string makes an angle \(\theta\) with the downward vertical through \(O\) and the string is still taut, the tension in the string is \(T\).

1. Show that \(T = m g ( 3 \cos \theta + 2 )\) At the instant when the particle reaches the point \(B\), the string becomes slack.
2. Find the speed of \(P\) at \(B\).
3. Find the greatest height above \(O\) reached by \(P\) in the subsequent motion.

5. \begin{figure}[h] \end{figure} 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 particle \(P\) is held at rest vertically below \(O\), with the string taut, as shown in Figure 4. The particle is then projected horizontally with speed \(u\), where \(u > \sqrt { 2 a g }\)
Air resistance is modelled as being negligible.
At the instant when the string makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\) and the string goes slack.

1. Show that \(3 v ^ { 2 } = u ^ { 2 } - 2 a g\) From the instant when the string goes slack to the instant when \(O P\) is next horizontal, \(P\) moves as a projectile. The time from the instant when the string goes slack to the instant when \(O P\) is next horizontal is \(T\) Given that \(\theta = 30 ^ { \circ }\)
2. show that \(T = \frac { 2 v } { g }\)
3. Hence, show that the string goes taut again when it is next horizontal.

6. \begin{figure}[h] \end{figure} 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 particle is held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal, as shown in Figure 4. The particle is projected vertically downwards with speed \(\sqrt { \frac { 9 a g } { 5 } }\)
When the string makes an angle \(\theta\) with the downward vertical through \(O\) and the string is still taut, the tension in the string is \(S\).

1. Show that \(S = \frac { 3 } { 5 } m g ( 5 \cos \theta + 3 )\) At the instant when the string becomes slack, the speed of \(P\) is \(v\)
2. Show that \(v = \sqrt { \frac { 3 a g } { 5 } }\)
3. Find the maximum height of \(P\) above the horizontal level of \(O\)

6. \begin{figure}[h] \end{figure} 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 particle \(P\) is held at rest with the string taut and horizontal and is then projected vertically downwards with speed \(u\), as shown in Figure 5. Air resistance is modelled as being negligible.
At the instant when the string has turned through an angle \(\theta\) and the string is taut, the tension in the string is \(T\).

1. Show that \(T = \frac { m u ^ { 2 } } { a } + 3 m g \sin \theta\) Given that \(u = 2 \sqrt { \frac { 3 a g } { 5 } }\)
2. find, in terms of \(a\) and \(g\), the speed of \(P\) at the instant when the string goes slack.
3. Hence find, in terms of \(a\), the maximum height of \(P\) above \(O\) in the subsequent motion.

4. \begin{figure}[h] \end{figure} 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 point \(O\). The point \(A\) is vertically below \(O\), and \(O A = a\). The particle is projected horizontally from \(A\) with speed \(\sqrt { } ( 3 a g )\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) and the string is still taut, the tension in the string is \(T\) and the speed of \(P\) is \(v\), as shown in Figure 2.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Show that \(T = ( 1 - 3 \cos \theta ) m g\). The string becomes slack when \(P\) is at the point \(B\).
3. Find, in terms of \(a\), the vertical height of \(B\) above \(A\). After the string becomes slack, the highest point reached by \(P\) is \(C\).
4. Find, in terms of \(a\), the vertical height of \(C\) above \(B\).

7. \begin{figure}[h] \end{figure} 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 fixed at a point \(O\). The particle is held with the string taut and \(O P\) horizontal. It is then projected vertically downwards with speed \(u\), where \(u ^ { 2 } = \frac { 3 } { 2 } g a\). When \(O P\) has turned through an angle \(\theta\) and the string is still taut, the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Fig. 3.

1. Find an expression for \(v ^ { 2 }\) in terms of \(a , g\) and \(\theta\).
2. Find an expression for \(T\) in terms of \(m , g\) and \(\theta\).
3. Prove that the string becomes slack when \(\theta = 210 ^ { \circ }\).
4. State, with a reason, whether \(P\) would complete a vertical circle if the string were replaced by a light rod. After the string becomes slack, \(P\) moves freely under gravity and is at the same level as \(O\) when it is at the point \(A\).
5. Explain briefly why the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 2 } g a \right)\). The direction of motion of \(P\) at \(A\) makes an angle \(\varphi\) with the horizontal.
6. Find \(\varphi\).

6. One end of a light inextensible string of length \(l\) is attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\) which is hanging freely at rest at point \(B\). The particle \(P\) is projected horizontally from \(B\) with speed \(\sqrt { } ( 3 g l )\). When \(A P\) makes an angle \(\theta\) with the downward vertical and the string remains taut, the tension in the string is \(T\).

1. Show that \(T = m g ( 1 + 3 \cos \theta )\).
2. Find the speed of \(P\) at the instant when the string becomes slack.
3. Find the maximum height above the level of \(B\) reached by \(P\).

4. \begin{figure}[h] \end{figure} 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 point \(O\). The point \(A\) is vertically below \(O\), and \(O A = a\). The particle is projected horizontally from \(A\) with speed \(\sqrt { } ( 3 a g )\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) and the string is still taut, the tension in the string is \(T\) and the speed of \(P\) is \(v\), as shown in Figure 2.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Show that \(T = ( 1 - 3 \cos \theta ) m g\). The string becomes slack when \(P\) is at the point \(B\).
3. Find, in terms of \(a\), the vertical height of \(B\) above \(A\). After the string becomes slack, the highest point reached by \(P\) is \(C\).
4. Find, in terms of \(a\), the vertical height of \(C\) above \(B\).

6. \begin{figure}[h] \end{figure} 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 \(O\). At time \(t = 0 , P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 5 } { 2 } g a \right)\) from a point \(A\) which is at the same level as \(O\) and a distance \(a\) from \(O\). When the string has turned through an angle \(\theta\) and the string is still taut, the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 2.

1. Show that \(v ^ { 2 } = \frac { g a } { 2 } ( 5 + 4 \sin \theta )\).
2. Find \(T\) in terms of \(m , g\) and \(\theta\). The string becomes slack when \(\theta = \alpha\).
3. Find the value of \(\alpha\). The particle is projected again from \(A\) with the same velocity as before. When \(P\) is at the same level as \(O\) for the first time after leaving \(A\), the string meets a small smooth peg \(B\) which has been fixed at a distance \(\frac { 1 } { 2 } a\) from \(O\). The particle now moves on an arc of a circle centre \(B\). Given that the particle reaches the point \(C\), which is \(\frac { 1 } { 2 } a\) vertically above the point \(B\), without the string going slack,
4. find the tension in the string when \(P\) is at the point \(C\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(5 m\) 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 particle is held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal, as shown in Figure 6. The particle is projected vertically downwards with speed \(\sqrt { } \left( \frac { 9 a g } { 5 } \right)\). When the string makes an angle \(\theta\) with the downward vertical through \(O\) and the string is still taut, the tension in the string is \(T\).

1. Show that \(T = 3 m g ( 5 \cos \theta + 3 )\). At the instant when the particle reaches the point \(B\) the string becomes slack.
2. Find the speed of \(P\) at \(B\). At time \(t = 0 , P\) is at \(B\). At time \(t\), before the string becomes taut once more, the coordinates of \(P\) are \(( x , y )\) referred to horizontal and vertical axes with origin \(O\). The \(x\)-axis is directed along \(O A\) produced and the \(y\)-axis is vertically upward.
3. Find
   1. \(x\) in terms of \(t , a\) and \(g\),
   2. \(y\) in terms of \(t , a\) and \(g\).

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 particle is held so that the string is taut, with \(O P\) horizontal. The particle is projected downwards with speed \(\sqrt { } \left( \frac { 2 } { 5 } a g \right)\) and begins to move in a vertical circle. The string breaks when its tension is equal to \(\frac { 11 } { 5 } m g\).

1. Show that the string breaks when \(O P\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\cos \theta = \frac { 3 } { 5 }\). Find the speed of \(P\) at this instant.
2. For the subsequent motion after the string breaks, find the distance \(O P\) when the particle \(P\) is vertically below \(O\).

1 A particle of mass \(m\) is attached to one end \(A\) of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and the particle hangs in equilibrium under gravity. The particle is projected horizontally so that it starts to move in a vertical circle. The string slackens after turning through an angle of \(120 ^ { \circ }\). Show that the speed of the particle is then \(\sqrt { } \left( \frac { 1 } { 2 } g a \right)\) and find the initial speed of projection.

4 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 \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\). Show that the least possible value of \(u\) is \(\sqrt { } ( a g )\). It is now given that \(u = \sqrt { } ( \mathrm { ag } )\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac { 1 } { 4 } m\). Find the speed of the combined particle immediately after the collision. In the subsequent motion, when \(O P\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m , g\) and \(\theta\). Find the value of \(\cos \theta\) when the string becomes slack.

4 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 \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\).

1. Show that the least possible value of \(u\) is \(\sqrt { } ( a g )\).
   It is now given that \(u = \sqrt { } ( a g )\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac { 1 } { 4 } m\).
2. Find the speed of the combined particle immediately after the collision.
   In the subsequent motion, when \(O P\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\).
3. Find an expression for \(T\) in terms of \(m , g\) and \(\theta\).
4. Find the value of \(\cos \theta\) when the string becomes slack.

3 One end of a light inextensible string of length 0.75 m is attached to a particle \(A\) of mass 2.8 kg . The other end of the string is attached to a fixed point \(O\). \(A\) is projected horizontally with speed \(6 \mathrm {~ms} ^ { - 1 }\) from a point 0.75 m vertically above \(O\) (see Fig. 3). When \(O A\) makes an angle \(\theta\) with the upward vertical the speed of \(A\) is \(v \mathrm {~ms} ^ { - 1 }\). \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 50.7 - 14.7 \cos \theta\).
2. Given that the string breaks when the tension in it reaches 200 N , find the angle that \(O A\) turns through between the instant that \(A\) is projected and the instant that the string breaks.

6 A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string of length 0.8 m . The other end of the string is attached to a fixed point \(O . P\) is at rest vertically below \(O\) when it experiences a horizontal impulse of magnitude 20 Ns . In the subsequent motion the angle the string makes with the downwards vertical through \(O\) is denoted by \(\theta\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-4_387_502_1434_255}

1. Find the magnitude of the acceleration of \(P\) at the first instant when \(\theta = \frac { 1 } { 3 } \pi\) radians.
2. Determine the value of \(\theta\) at which the string first becomes slack.

1. A particle \(P\) 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 \(O\). When \(P\) is hanging at rest vertically below \(O\), it is given a horizontal speed \(u \mathrm {~ms} ^ { - 1 }\) and starts to move in a vertical circle.

Given that the string becomes slack when it makes an angle of \(120 ^ { \circ }\) with the downward vertical through \(O\),

1. show that \(u ^ { 2 } = \frac { 7 g l } { 2 }\).
2. Find, in terms of \(l\), the greatest height above \(O\) reached by \(P\) in the subsequent motion.
   (7 marks)

6. A particle \(P\) of mass 0.4 kg hangs by a light, inextensible string of length 20 cm whose other end is attached to a fixed point \(O\). It is given a horizontal velocity of \(1.4 \mathrm {~ms} ^ { - 1 }\) so that it begins to move in a vertical circle. If in the ensuing motion the string makes an angle of \(\theta\) with the downward vertical through \(O\), show that

1. \(\theta\) cannot exceed \(60 ^ { \circ }\),
2. the tension, \(T \mathrm {~N}\), in the string is given by \(T = 3.92 ( 3 \cos \theta - 1 )\). If the string breaks when \(\cos \theta = \frac { 3 } { 5 }\) and \(P\) is ascending,
3. find the greatest height reached by \(P\) above the initial point of projection.

6
\includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of \(60 ^ { \circ }\) to the upward vertical, \(P\) is projected with speed \(2 \mathrm {~ms} ^ { - 1 }\) (see diagram). \(P\) begins to move without air resistance in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the upward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 8.9 - 9.8 \cos \theta\).
2. Find the tension in the string in terms of \(\theta\).
3. \(P\) does not move in a complete circle. Calculate the angle through which \(O P\) turns before \(P\) leaves the circular path.