Vertical circle: string becomes slack

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 \(O\).

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.

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.

6 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_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.

6 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-4_497_524_276_804} A particle \(P\) of mass 0.4 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 \(O\). With the string taut the particle is travelling in a circular path in a vertical plane. The angle between the string and the downward vertical is \(\theta ^ { \circ }\) (see diagram). When \(\theta = 0\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. At the instant when the string is horizontal, find the speed of \(P\) and the tension in the string.
2. At the instant when the string becomes slack, find the value of \(\theta\).

3 A particle P of mass 0.6 kg is connected to a fixed point O by a light inextensible string of length 1.25 m . When it is 1.25 m vertically below \(\mathrm { O } , \mathrm { P }\) is set in motion with horizontal velocity \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then moves in part of a vertical circle with centre O and radius 1.25 m . When OP makes an angle \(\theta\) with the downward vertical, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 3.1. \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 11.5 + 24.5 \cos \theta\).
2. Find the tension in the string in terms of \(\theta\).
3. Find the speed of P at the instant when the string becomes slack. A second light inextensible string, of length 0.35 m , is attached to P , and the other end of this string is attached to a point C which is 1.2 m vertically below O . The particle P now moves in a horizontal circle with centre C and radius 0.35 m , as shown in Fig. 3.2. The speed of P is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-3_518_488_1701_826} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
4. Find the tension in the string OP and the tension in the string CP.

3. \begin{figure}[h] \end{figure} A particle of mass \(m\) is suspended at a point \(A\) vertically below a fixed point \(O\) by a light inextensible string of length \(a\) as shown in Figure 2. The particle is given a horizontal velocity \(u\) and subsequently moves along a circular arc until it reaches the point \(B\) where the string becomes slack. Given that the point \(B\) is at a height \(\frac { 1 } { 2 } a\) above the level of \(O\),

1. show that \(\angle B O A = 120 ^ { \circ }\),
2. show that \(u ^ { 2 } = \frac { 7 } { 2 } g a\).

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 with the string taut and \(O P\) horizontal. The particle is then projected vertically downwards with speed \(u\), where \(u ^ { 2 } = \frac { 9 } { 5 } \mathrm { gl }\). When \(O P\) has turned through an angle \(\alpha\) and the string is still taut, the speed of \(P\) is \(v\), as shown in Figure 5. At this instant the tension in the string is \(T\).

1. Show that \(T = 3 m g \sin \alpha + \frac { 9 } { 5 } m g\)
2. Find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the string goes slack.
3. Find, in terms of \(l\), the greatest vertical height reached by \(P\) above the level of \(O\).

10 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 free end of the string and the particle hangs at rest vertically below \(O\). The particle is projected horizontally with speed \(u\).

1. Find the tension in the string when it makes an angle \(\theta\) with the downward vertical, whilst the string remains taut.
2. Deduce that the particle will perform complete circles provided that \(u ^ { 2 } \geqslant 5 a g\).
3. It is given that \(u ^ { 2 } = 4 a g\). Find
   1. the tension in the string when \(\theta = 60 ^ { \circ }\),
   2. the value of \(\theta\), to the nearest degree, at the instant when the string becomes slack.

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 \(\sqrt{(ag)}\). [2] It is now given that \(u = \sqrt{(ag)}\). 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. [4] In the subsequent motion, when \(OP\) 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 \(\theta\). [5] Find the value of \(\cos \theta\) when the string becomes slack. [2]

\includegraphics{figure_3} A particle 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 point \(A\) is such that \(OA = a\) and \(OA\) makes an angle \(\alpha\) with the upward vertical, where \(\tan \alpha = \frac{15}{8}\). The particle is projected downwards from \(A\) with speed \(u\) perpendicular to the string and moves in a vertical plane (see diagram). The string becomes slack after the string has rotated through \(270°\) from its initial position, with the particle now at the point \(B\). \begin{enumerate}[label=(\roman*)] \item Show that \(u^2 = 2ag\). [5] \item Find the maximum tension in the string as the particle moves from \(A\) to \(B\). [4] \end{enumerate]