Two strings, two fixed points

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. }

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 \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-3_604_490_258_831} A small ball \(B\) of mass 0.5 kg is attached to points \(P\) and \(Q\) on a fixed vertical axis by two light inextensible strings of equal length. Both of the strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical, as shown in the diagram. The ball moves with constant speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.8 m . Find the tension in the string \(P B\).

5 A small ball \(B\) of mass 0.2 kg is attached to fixed points \(P\) and \(Q\) by two light inextensible strings of equal length. \(P\) is vertically above \(Q\), the strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical. \(B\) moves with constant speed in a horizontal circle of radius 0.6 m .

1. Given that the tension in the string \(P B\) is 7 N , calculate
   1. the tension in string \(Q B\),
   2. the speed of \(B\).
   3. Given instead that \(B\) is moving with angular speed \(7 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the tension in the string \(Q B\).

5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(O A\) is a line of greatest slope of the plane with \(A\) below the level of \(O\) and \(O A = 0.8 \mathrm {~m}\). The particle \(P\) is released from rest at \(A\).

1. Find the initial acceleration of \(P\).
2. Find the greatest speed of \(P\). \(6 \quad A\) and \(B\) are two fixed points on a vertical axis with \(A 0.6 \mathrm {~m}\) above \(B\). A particle \(P\) of mass 0.3 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by a light elastic string with modulus of elasticity 46 N . The particle \(P\) moves with constant angular speed \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre at the mid-point of \(A B\).
3. Find the speed of \(P\).
4. Calculate the tension in the string \(B P\) and hence find the natural length of this string. \includegraphics[max width=\textwidth, alt={}, center]{4cd525d5-d59b-4ab9-85a3-fc3d97fd09fc-10_540_574_260_781} \(A B C\) is the cross-section through the centre of mass of a uniform prism which rests with \(A B\) on a rough horizontal surface. \(A B = 0.4 \mathrm {~m}\) and \(C\) is 0.9 m above the surface (see diagram). The prism is on the point of toppling about its edge through \(B\).
5. Show that angle \(B A C = 48.4 ^ { \circ }\), correct to 3 significant figures.
   A force of magnitude 18 N acting in the plane of the cross-section and perpendicular to \(A C\) is now applied to the prism at \(C\). The prism is on the point of rotating about its edge through \(A\).
6. Calculate the weight of the prism.
7. Given also that the prism is on the point of slipping, calculate the coefficient of friction between the prism and the surface.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-06_671_504_255_824} \(A\) and \(B\) are two fixed points on a vertical axis with \(A\) above \(B\). A particle \(P\) of mass 0.4 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by another light inextensible string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) between \(A\) and \(B\). Angle \(B A P = 30 ^ { \circ }\) and angle \(A B P = 70 ^ { \circ }\) (see diagram).

1. Given that the tensions in the two strings are equal, find the speed of \(P\).
2. Given instead that the angular speed of \(P\) is \(12 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tensions in the strings.

5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(O A\) is a line of greatest slope of the plane with \(A\) below the level of \(O\) and \(O A = 0.8 \mathrm {~m}\). The particle \(P\) is released from rest at \(A\).

1. Find the initial acceleration of \(P\).
2. Find the greatest speed of \(P\). \(6 \quad A\) and \(B\) are two fixed points on a vertical axis with \(A 0.6 \mathrm {~m}\) above \(B\). A particle \(P\) of mass 0.3 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by a light elastic string with modulus of elasticity 46 N . The particle \(P\) moves with constant angular speed \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre at the mid-point of \(A B\).
3. Find the speed of \(P\).
4. Calculate the tension in the string \(B P\) and hence find the natural length of this string. \includegraphics[max width=\textwidth, alt={}, center]{42de91da-d65e-40e7-8de5-f40eda927850-10_540_574_260_781} \(A B C\) is the cross-section through the centre of mass of a uniform prism which rests with \(A B\) on a rough horizontal surface. \(A B = 0.4 \mathrm {~m}\) and \(C\) is 0.9 m above the surface (see diagram). The prism is on the point of toppling about its edge through \(B\).
5. Show that angle \(B A C = 48.4 ^ { \circ }\), correct to 3 significant figures.
   A force of magnitude 18 N acting in the plane of the cross-section and perpendicular to \(A C\) is now applied to the prism at \(C\). The prism is on the point of rotating about its edge through \(A\).
6. Calculate the weight of the prism.
7. Given also that the prism is on the point of slipping, calculate the coefficient of friction between the prism and the surface.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6. \begin{figure}[h] \end{figure} A light inextensible string of length \(14 a\) has its ends attached to two fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\) and \(A B = 10 a\). A particle of mass \(m\) is attached to the string at the point \(P\), where \(A P = 8 a\). The particle moves in a horizontal circle with constant angular speed \(\omega\) and with both parts of the string taut, as shown in Figure 3.

1. Show that angle \(A P B = 90 ^ { \circ }\).
2. Show that the time for the particle to make one complete revolution is less than $$2 \pi \sqrt { \left( \frac { 32 a } { 5 g } \right) } .$$

3. \begin{figure}[h] \end{figure} A light inextensible string has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(m\). An identical string has one end attached to the fixed point \(B\), where \(B\) is vertically below \(A\) and \(A B = 4 a\), and the other end attached to \(P\), as shown in Figure 2. The particle is moving in a horizontal circle with constant angular speed \(\omega\), with both strings taut and inclined at \(30 ^ { \circ }\) to the vertical. The tension in the upper string is twice the tension in the lower string. Find \(\omega\) in terms of \(a\) and \(g\).

4. \begin{figure}[h] \end{figure} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\) and \(A B = 7 a\). A particle \(P\) of mass \(m\) is fixed to the string and moves with constant angular speed \(\omega\) in a horizontal circle of radius \(4 a\). The centre of the circle is \(C\), where \(C\) lies on \(A B\) and \(A C = 3 a\), as shown in Figure 3. Both parts of the string are taut.

1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 4 a \omega ^ { 2 } + g \right)\).
2. Find the tension in \(B P\).
3. Deduce that \(\omega \geqslant \sqrt { \frac { g } { k a } }\), stating the value of \(k\).

2. \begin{figure}[h] \end{figure} A small ball \(P\) of mass \(m\) is attached to the midpoint of a light inextensible string of length \(2 a\). The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\) and \(A B = a\), as shown in Figure 1. The system rotates about the line \(A B\) with constant angular speed \(\omega\). The ball moves in a horizontal circle with both parts of the string taut. The tension in the string must be less than \(3 m g\) otherwise the string will break. Given that the time taken by the ball to complete one revolution is \(S\), show that $$\pi \sqrt { \frac { a } { g } } < S < \pi \sqrt { \frac { k a } { g } }$$ stating the value of the constant \(k\).

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 3 kg is attached by two light inextensible strings to two fixed points \(A\) and \(B\) on a fixed vertical pole. Both strings are taut and \(P\) is moving in a horizontal circle with constant angular speed \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). String \(A P\) is inclined at \(30 ^ { \circ }\) to the vertical. String \(B P\) has length 0.4 m and \(A\) is 0.4 m vertically above \(B\), as shown in Figure 2 . Find the tension in

1. \(A P\),
2. \(B P\).

3. \begin{figure}[h] \end{figure} A light inextensible string of length \(7 l\) has one end attached to a fixed point \(A\) and the other end attached to a fixed point \(B\), where \(A\) is vertically above \(B\) and \(A B = 5\) l. A particle of mass \(m\) is attached to the string at the point \(C\) where \(A C = 4 l\), as shown in Figure 1. The particle moves in a horizontal circle with constant angular speed \(\omega\). Both parts of the string are taut.

1. Find, in terms of \(m , g , l\) and \(\omega\),
   1. the tension in \(A C\),
   2. the tension in \(B C\). The time taken by the particle to complete one revolution is \(R\).
      Given that \(R \leqslant k \pi \sqrt { \frac { l } { 5 g } }\)
2. find the least possible value of \(k\).

3. \begin{figure}[h] \end{figure} A small ball \(P\) of mass \(m\) is attached to the midpoint of a light inextensible string of length \(4 l\). The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). Both strings are taut and \(A P\) makes an angle of \(30 ^ { \circ }\) with \(A B\), as shown in Figure 1. The ball is moving in a horizontal circle with constant angular speed \(\omega\).

1. Find, in terms of \(m , g , l\) and \(\omega\),
   1. the tension in \(A P\),
   2. the tension in \(B P\).
2. Show that \(\omega ^ { 2 } \geqslant \frac { g \sqrt { 3 } } { 3 l }\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to the ends of two light inextensible strings. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(B\) is vertically below \(A\) and \(A B = l\). The particle is moving with constant angular speed \(\omega\) in a horizontal circle. Both strings are taut and inclined at \(30 ^ { \circ }\) to \(A B\), as shown in Figure 3.

1. 1. Show that the tension in \(A P\) is \(\frac { m \sqrt { 3 } } { 6 } \left( 2 g + l \omega ^ { 2 } \right)\)
   2. Find the tension in \(B P\).
2. Show that the time taken by \(P\) to complete one revolution is less than \(\pi \sqrt { \frac { 2 l } { g } }\)

5 \includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-3_729_739_868_703} A particle \(P\) of mass 0.2 kg is attached to one end of each of two light inextensible strings, one of length 0.4 m and one of length 0.3 m . The other end of the longer string is attached to a fixed point \(A\), and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). The particle moves in a horizontal circle of radius 0.24 m at a constant angular speed of \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Both strings are taut, the tension in \(A P\) is \(S \mathrm {~N}\) and the tension in \(B P\) is \(T \mathrm {~N}\).

1. By resolving vertically, show that \(4 S = 3 T + 9.8\).
2. Find another equation connecting \(S\) and \(T\) and hence calculate the tensions, correct to 1 decimal place. \section*{[Questions 6 and 7 are printed overleaf.]}

7 Two identical light, inextensible strings \(S _ { 1 }\) and \(S _ { 2 }\) are each of length 5 m . Two identical particles \(P\) and \(Q\) are each of mass 1.5 kg . One end of \(S _ { 1 }\) is attached to \(P\). The other end of \(S _ { 1 }\) is attached to a fixed point \(A\) on a smooth horizontal plane. \(P\) moves with constant speed in a horizontal circular path with \(A\) as its centre (see Fig. 1). One end of \(S _ { 2 }\) is attached to \(Q\). The other end of \(S _ { 2 }\) is attached to a fixed point \(B\). \(Q\) moves with constant speed in a horizontal circular path around a point \(O\) which is vertically below \(B\). At any instant, \(B Q\) makes an angle of \(\theta\) with the downward vertical through \(B\) (see Fig. 2). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Given that the angular speed of \(P\) is the same as the angular speed of \(Q\), show that the tensions in \(S _ { 1 }\) and \(S _ { 2 }\) have the same magnitude.
2. You are given instead that the kinetic energy of \(P\) is 39.2 J and that this is the same as the kinetic energy of \(Q\). Determine the difference between the times taken by \(P\) and \(Q\) to complete one revolution. Give your answer in an exact form.

5 Two light inextensible strings, of lengths 0.4 m and 0.2 m , each have one end attached to a particle, \(P\), of mass 4 kg . The other ends of the strings are attached to the points \(A\) and \(B\) respectively. The point \(A\) is vertically above the point \(B\). The particle moves in a horizontal circle, centre \(B\) and radius 0.2 m , at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle and strings are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_396_558_587_735} $$\text { ← } 0.2 \mathrm {~m} \longrightarrow$$

1. Calculate the magnitude of the acceleration of the particle.
2. Show that the tension in string \(P A\) is 45.3 N , correct to three significant figures.
3. Find the tension in string \(P B\).

7 Two light inextensible strings each have one end attached to a particle, \(P\), of mass 4 kg . The other ends of the strings are attached to the fixed points \(A\) and \(B\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed in a horizontal circle. The centre, \(C\), of this circle is directly below the point \(B\). The two strings are inclined at \(30 ^ { \circ }\) and \(50 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut. As the particle moves in the horizontal circle, the tension in the string \(B P\) is 20 N . \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-5_750_469_742_781}

1. Find the tension in the string \(A P\).
2. The speed of the particle is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the length of \(C P\), the radius of the horizontal circle.

4 A particle, \(P\), of mass 5 kg is attached to two light inextensible strings, \(A P\) and \(B P\). The other ends of the strings are attached to the fixed points \(A\) and \(B\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in a horizontal circle of radius 0.6 metres with centre \(B\). The string \(A P\) is inclined at \(20 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut when the particle is moving. \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-08_835_568_568_719}

1. Find the tension in the string \(A P\).
2. The speed of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the tension, \(T _ { B P }\), in the string \(B P\) is given by $$T _ { B P } = \frac { 25 } { 3 } v ^ { 2 } - 5 g \tan 20 ^ { \circ }$$
3. Find \(v\) when the tensions in the two strings are equal.

6 \includegraphics[max width=\textwidth, alt={}, center]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-3_670_613_274_767} A particle \(P\) of mass 0.3 kg is attached to one end of each of two light inextensible strings. The other end of the longer string is attached to a fixed point \(A\) and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.4 m long. \(P B\) makes an angle of \(60 ^ { \circ }\) with the vertical. The particle moves in a horizontal circle with constant angular speed and with both strings taut (see diagram). The tension in the string \(A P\) is 5 N . Calculate

1. the tension in the string \(P B\),
2. the angular speed of \(P\),
3. the kinetic energy of \(P\).

6 \includegraphics[max width=\textwidth, alt={}, center]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-4_794_735_264_705} A particle \(P\) of mass 0.5 kg is attached to points \(A\) and \(B\) on a fixed vertical axis by two light inextensible strings of equal length. Both strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical (see diagram). The particle moves with constant speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.4 m .

1. Calculate the tensions in the two strings. The particle now moves with constant angular speed \(\omega\) rad s \(^ { - 1 }\) and the string \(B P\) is on the point of becoming slack.
2. Calculate \(\omega\).

5 A particle \(P\), of mass 2 kg , is attached to fixed points \(A\) and \(B\) by light inextensible strings, each of length 2 m . \(A\) and \(B\) are 3.2 m apart with \(A\) vertically above \(B\). The particle \(P\) moves in a horizontal circle with centre at the mid-point of \(A B\).

1. Find the tension in each string when the angular speed of \(P\) is \(4 \mathrm { rads } ^ { - 1 }\).
2. Find the least possible speed of \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-4_314_757_285_708} A ball of mass 0.08 kg is attached by two strings to a fixed vertical post. The strings have lengths 2.5 m and 2.4 m , as shown in the diagram. The ball moves in a horizontal circle, of radius 2.4 m , with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Each string is taut and the lower string is horizontal. The modelling assumptions made are that both strings are light and inextensible, and that there is no air resistance.

1. Find the tension in each string when \(v = 10.5\).
2. Find the least value of \(v\) for which the lower string is taut.

2 A light inextensible string of length 5 m has one end attached to a fixed point A and the other end attached to a particle P of mass 0.72 kg . At first, P is moving in a vertical circle with centre A and radius 5 m . When P is at the highest point of the circle it has speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the tension in the string when the speed of P is \(15 \mathrm {~ms} ^ { - 1 }\). The particle P now moves at constant speed in a horizontal circle with radius 1.4 m and centre at the point C which is 4.8 m vertically below A .
2. Find the tension in the string.
3. Find the time taken for P to make one complete revolution. Another light inextensible string, also of length 5 m , now has one end attached to P and the other end attached to the fixed point B which is 9.6 m vertically below A . The particle P then moves with constant speed \(7 \mathrm {~ms} ^ { - 1 }\) in the circle with centre C and radius 1.4 m , as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86d79489-aec1-4c94-bef6-45b007f818a0-3_693_465_1078_817} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
4. Find the tension in the string PA and the tension in the string PB .