Two springs/strings system equilibrium

7. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) are 4.2 m apart on a smooth horizontal floor. One end of a light elastic spring, of natural length 1.8 m and modulus of elasticity 20 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic spring, of natural length 0.9 m and modulus of elasticity 15 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 5.

1. Show that \(A O = 2.7 \mathrm {~m}\). The particle \(P\) now receives an impulse acting in the direction \(O B\) and moves away from \(O\) towards \(B\). In the subsequent motion \(P\) does not reach \(B\).
2. Show that \(P\) moves with simple harmonic motion about centre \(O\). The mass of \(P\) is 10 kg and the magnitude of the impulse is \(J \mathrm { Ns }\). Given that \(P\) first comes to instantaneous rest at the point \(C\) where \(A C = 2.9 \mathrm {~m}\),
3. 1. find the value of \(J\),
   2. find the time taken by \(P\) to travel a total distance of 0.5 m from when it first leaves \(O\).

7. \begin{figure}[h] \end{figure} Two points \(A\) and \(B\) lie on a smooth horizontal table where \(A B = 41\).
A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length I and modulus of elasticity 2 mg . The other end of the spring is attached to A . The particle P is also attached to one end of another light elastic spring of natural length I and modulus of elasticity mg . The other end of the spring is attached to B.
The particle \(P\) rests in equilibrium on the table at the point 0 , where \(A 0 = \frac { 5 } { 3 } I\), as shown in Figure 7.
The particle \(P\) is moved a distance \(\frac { 1 } { 2 } \mathrm { I }\) along the table, from 0 towards \(A\), and released from rest.

1. Show that P moves with simple harmonic motion of period T , where $$\mathrm { T } = 2 \pi \sqrt { \frac { l } { 3 g } }$$
2. Find, in terms of I and g , the speed of P as it passes through 0 .
3. Find, in terms of g , the maximum acceleration of P .
4. Find the exact time, in terms of I and g , from the instant when P is released from rest to the instant when P is first moving with speed \(\frac { 3 } { 4 } \sqrt { g l }\)
   \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-20_2269_56_311_1978} \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
   VILU SIHIL NI GLIUM ION OC
   VEYV SIHI NI ELIUM ION OC

7. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) are 7 m apart on a smooth horizontal surface.
A light elastic string has natural length 2 m and modulus of elasticity 4 N . One end of the string is attached to a particle \(P\) of mass 2 kg and the other end is attached to \(A\) Another light elastic string has natural length 3 m and modulus of elasticity 2 N . One end of this string is attached to \(P\) and the other end is attached to \(B\) The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 4.

1. Show that \(O A = 2.5 \mathrm {~m}\). The particle \(P\) now receives an impulse of magnitude 6Ns in the direction \(O B\)
2. 1. Show that \(P\) initially moves with simple harmonic motion with centre \(O\)
   2. Determine the amplitude of this simple harmonic motion. The point \(C\) lies on \(O B\). As \(P\) passes through \(C\) the string attached to \(B\) becomes slack.
3. Find the speed of \(P\) as it passes through \(C\)
4. Find the time taken for \(P\) to travel directly from \(O\) to \(C\)

7. \begin{figure}[h] \end{figure} Figure 5 shows two fixed points, \(A\) and \(B\), which are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 1.25 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to \(A\) A second light elastic string, of natural length 1.2 m and modulus of elasticity \(\lambda\) newtons, has one end attached to \(P\) and the other end attached to \(B\) Initially \(P\) rests in equilibrium at the point \(O\), where \(A O = 3 \mathrm {~m}\)

1. Show that \(\lambda = 15\) The particle is now projected along the floor towards \(B\)
   At time \(t\) seconds, \(P\) is a displacement \(x\) metres from \(O\) in the direction \(O B\)
2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion where \(\ddot { x } = - 18 x\) The initial speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. Both strings are taut for \(T\) seconds during one complete oscillation.
4. Find the value of \(T\)

7. \begin{figure}[h] \end{figure} \(A\) and \(B\) are two points on a smooth horizontal floor, where \(A B = 5 \mathrm {~m}\). A particle \(P\) has mass 0.5 kg . One end of a light elastic spring, of natural length 2 m and modulus of elasticity 16 N , is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length 1 m and modulus of elasticity 12 N , are attached to \(P\) and \(B\), as shown in Figure 4.

1. Find the extensions in the two springs when the particle is at rest in equilibrium. Initially \(P\) is at rest in equilibrium. It is then set in motion and starts to move towards \(B\). In the subsequent motion \(P\) does not reach \(A\) or \(B\).
2. Show that \(P\) oscillates with simple harmonic motion about the equilibrium position.
3. Given that the initial speed of \(P\) is \(\sqrt { } 10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the proportion of time in each complete oscillation for which \(P\) stays within 0.25 m of the equilibrium position.

6. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) are 3.75 m apart on a smooth horizontal floor. A particle \(P\) has mass 0.8 kg . One end of a light elastic spring, of natural length 1.5 m and modulus of elasticity 24 N , is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length 0.75 m and modulus of elasticity 18 N , are attached to \(P\) and \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 5.

1. Show that \(A O = 2.4 \mathrm {~m}\). The point \(C\) lies on the straight line \(A O B\) between \(O\) and \(B\). The particle \(P\) is held at \(C\) and released from rest.
2. Show that \(P\) moves with simple harmonic motion. The maximum speed of \(P\) is \(\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the time taken by \(P\) to travel 0.3 m from \(C\).

6. \begin{figure}[h] \end{figure} Two points \(A\) and \(B\) are 6 m apart on a smooth horizontal floor. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2.5 m and modulus of elasticity 20 N . The other end of the spring is attached to \(A\). A second light elastic spring, of natural length 1.5 m and modulus of elasticity 18 N , has one end attached to \(P\) and the other end attached to \(B\), as shown in Figure 3. Initially \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line.

1. Find the length of \(A O\). The particle \(P\) now receives an impulse of magnitude 6 N s acting in the direction \(O B\) and \(P\) starts to move towards \(B\).
2. Show that \(P\) moves with simple harmonic motion about \(O\).
3. Find the amplitude of the motion.
4. Find the time taken by \(P\) to travel 1.2 m from \(O\).

7. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) are 4 m apart on a smooth horizontal floor. One end of a light elastic string, of natural length 1.8 m and modulus of elasticity 45 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic string, of natural length 1.2 m and modulus of elasticity 20 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 6.

1. Show that \(A O = 2.2 \mathrm {~m}\). The point \(C\) lies on the straight line \(A O B\) with \(A C = 2.7 \mathrm {~m}\). The mass of \(P\) is 0.6 kg . The particle \(P\) is held at \(C\) and then released from rest.
2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion with centre \(O\). The point \(D\) lies on the straight line \(A O B\) with \(A D = 1.8 \mathrm {~m}\). When \(P\) reaches \(D\) the string \(P B\) breaks.
3. Find the time taken by \(P\) to move directly from \(C\) to \(A\).

A particle of mass 0.1 kg lies on a smooth horizontal table on the line between two points \(A\) and \(B\) on the table, which are 6 m apart. The particle is joined to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 60 N , and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 20 N . The mid-point of \(A B\) is \(M\), and \(O\) is the point between \(M\) and \(B\) at which the particle can rest in equilibrium. Show that \(M O = 0.2 \mathrm {~m}\). The particle is held at \(M\) and then released. Show that the equation of motion is $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } = - 500 y$$ where \(y\) metres is the displacement from \(O\) in the direction \(O B\) at time \(t\) seconds, and state the period of the motion. For the instant when the particle is 0.3 m from \(M\) for the first time, find

1. the speed of the particle,
2. the time taken, after release, to reach this position.

5 A particle \(P\) of mass \(m\) lies on a smooth horizontal surface. \(A\) and \(B\) are fixed points on the surface, where \(A B = 10 a\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(8 m g\), joins \(P\) to \(A\). Another light elastic string, of natural length \(4 a\) and modulus of elasticity \(16 m g\), joins \(P\) to \(B\). Show that when \(P\) is in equilibrium, \(A P = 4 a\). The particle is held at rest at the point \(C\) between \(A\) and \(B\) on the line \(A B\) where \(A C = 3 a\). The particle is now released.

1. Show that the subsequent motion of \(P\) is simple harmonic with period \(\pi \sqrt { } \left( \frac { a } { 2 g } \right)\).
2. Find the maximum speed of \(P\).

5 The points \(A\) and \(B\) are on a smooth horizontal table at a distance \(8 a\) apart. A particle \(P\) of mass \(m\) lies on the table on the line \(A B\), between \(A\) and \(B\). The particle is attached to \(A\) by a light elastic string of natural length \(3 a\) and modulus of elasticity 6 mg , and to \(B\) by a light elastic string of natural length \(2 a\) and modulus of elasticity \(m g\). In equilibrium, \(P\) is at the point \(O\) on \(A B\).

1. Show that \(A O = 3.6 a\). The particle is released from rest at the point \(C\) on \(A B\), between \(A\) and \(B\), where \(A C = 3.4 a\).
2. Show that \(P\) moves in simple harmonic motion and state the period.
3. Find the greatest speed of \(P\).

5 The fixed points \(A\) and \(B\) are on a smooth horizontal surface with \(A B = 2.6 \mathrm {~m}\). One end of a light elastic spring, of natural length 1.25 m and modulus of elasticity \(\lambda \mathrm { N }\), is attached to \(A\). The other end is attached to a particle \(P\) of mass 0.4 kg . One end of a second light elastic spring, of natural length 1.0 m and modulus of elasticity \(0.6 \lambda \mathrm {~N}\), is attached to \(B\); its other end is attached to \(P\). The system is in equilibrium with \(P\) on the surface at the point \(E\).

1. Show that \(A E = 1.4 \mathrm {~m}\).
   The particle \(P\) is now displaced slightly from \(E\), along the line \(A B\).
2. Show that, in the subsequent motion, \(P\) performs simple harmonic motion.
3. Given that the period of the motion is \(\frac { 1 } { 7 } \pi \mathrm {~s}\), find the value of \(\lambda\).

The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac { 2 } { 3 } \mathrm {~kg}\) is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N . The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N , has one end attached to \(P\) and the other end attached to \(B\).

1. Show that when \(P\) is in equilibrium \(A P = 0.75 \mathrm {~m}\).
   The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.
2. Show that \(P\) performs simple harmonic motion and state the period of the motion.
3. Find the speed of \(P\) when it passes through the equilibrium position.
4. Find the speed of \(P\) when its acceleration is equal to half of its maximum value.

7
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-4_122_1009_265_571} As shown in the diagram, \(A\) and \(B\) are fixed points on a smooth horizontal table, where \(A B = 3 \mathrm {~m}\). A particle \(Q\) of mass 1.2 kg is attached to \(A\) by a light elastic string of natural length 1 m and modulus of elasticity \(180 \mathrm {~N} . Q\) is attached to \(B\) by a light elastic string of natural length 1.2 m and modulus of elasticity 360 N .

1. Verify that when \(Q\) is in equilibrium \(B Q = 1.5 \mathrm {~m}\).
   \(Q\) is projected towards \(B\) from the equilibrium position with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Subsequently \(Q\) oscillates with simple harmonic motion.
2. Show that the period of the motion is 0.314 s approximately.
3. Show that \(u \leqslant 6\).
4. Given that \(u = 6\), find the time taken for \(Q\) to move from the equilibrium position to a position 1.3 m from \(A\) for the first time.

3 A small block B has mass 2.5 kg . A light elastic string connects B to a fixed point P , and a second light elastic string connects \(B\) to a fixed point \(Q\), which is 6.5 m vertically below \(P\). The string PB has natural length 3.2 m and stiffness \(35 \mathrm { Nm } ^ { - 1 }\); the string BQ has natural length 1.8 m and stiffness \(5 \mathrm { Nm } ^ { - 1 }\). The block B is released from rest in the position 4.4 m vertically below P . You are given that B performs simple harmonic motion along part of the line PQ, and that both strings remain taut throughout the motion. Air resistance may be neglected. At time \(t\) seconds after release, the length of the string PB is \(x\) metres (see Fig. 3). \begin{figure}[h] \end{figure}

1. Find, in terms of \(x\), the tension in the string PB and the tension in the string BQ .
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 64 - 16 x\).
3. Find the value of \(x\) when B is at the centre of oscillation.
4. Find the period of oscillation.
5. Write down the amplitude of the motion and find the maximum speed of B.
6. Find the time after release when \(B\) is first moving downwards with speed \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 A light spring, with modulus of elasticity 686 N , has one end attached to a fixed point A . The other end is attached to a particle P of mass 18 kg which hangs in equilibrium when it is 2.2 m vertically below A .

1. Find the natural length of the spring AP . Another light spring has natural length 2.5 m and modulus of elasticity 145 N . One end of this spring is now attached to the particle P , and the other end is attached to a fixed point B which is 2.5 m vertically below P (so leaving the equilibrium position of P unchanged). While in its equilibrium position, P is set in motion with initial velocity \(3.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, as shown in Fig. 3. It now executes simple harmonic motion along part of the vertical line AB . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-4_721_383_726_831} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} At time \(t\) seconds after it is set in motion, P is \(x\) metres below its equilibrium position.
2. Show that the tension in the spring AP is \(( 176.4 + 392 x ) \mathrm { N }\), and write down an expression for the thrust in the spring BP.
3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 25 x\).
4. Find the period and the amplitude of the motion.
5. Find the magnitude and direction of the velocity of P when \(t = 2.4\).
6. Find the total distance travelled by P during the first 2.4 seconds of its motion.

6. \begin{figure}[h] \end{figure} Figure 2 shows a particle \(P\) of mass \(m\) which lies on a smooth horizontal table. It is attached to a point \(A\) on the table by a light elastic spring of natural length \(3 a\) and modulus of elasticity \(\lambda\), and to a point \(B\) on the table by a light elastic spring of natural length \(2 a\) and modulus of elasticity \(2 \lambda\). The distance between the points \(A\) and \(B\) is \(7 a\).

1. Show that in equilibrium \(A P = \frac { 9 } { 2 } a\). The particle is released from rest at a point \(Q\) where \(Q\) lies on the line \(A B\) and \(A Q = 5 a\).
2. Prove that the subsequent motion of the particle is simple harmonic with a period of \(\pi \sqrt { \frac { 3 m a } { \lambda } }\).
   (9 marks)

6. The diagram shows a particle \(P\), of mass 4 kg , lying on a smooth horizontal surface. It is attached by two light springs to fixed points \(A\) and \(B\), where \(A B = 2.8 \mathrm {~m}\).
Spring \(A P\) has natural length 0.8 m and modulus of elasticity 60 N .
Spring \(P B\) has natural length 1.2 m and modulus of elasticity 30 N .
\includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-5_231_1253_612_404} When \(P\) is in equilibrium, it is at the point \(C\).

1. Show that \(A C = 1 \mathrm {~m}\).
2. The particle \(P\) is pulled horizontally and is initially held at rest at the midpoint of \(A B\). The system is then released.
   1. Show that \(P\) performs Simple Harmonic Motion about centre \(C\) and find the period of its motion.
   2. Determine the shortest time taken for \(P\) to reach a position where there is no tension in the spring \(A P\). \section*{END OF PAPER}

13 Two light elastic strings each have one end attached to a particle \(B\) of mass \(3 c \mathrm {~kg}\), which rests on a smooth horizontal table. The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart.
\(A B C\) is a horizontal line.
\includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566} String \(A B\) has a natural length of 4 metres and a stiffness of \(5 c\) newtons per metre.
String \(B C\) has a natural length of 1 metre and a stiffness of \(c\) newtons per metre.
The particle is pulled a distance of \(\frac { 1 } { 3 }\) metre from its equilibrium position towards \(A\), and released from rest. 13

1. Show that the particle moves with simple harmonic motion.
   13
2. Find the speed of the particle when it is at a point \(P\), a distance \(\frac { 1 } { 4 }\) metre from the equilibrium position. Give your answer to two significant figures.
   [0pt] [4 marks]

15 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A particle \(P\) of mass \(m\) is attached to two light elastic strings, \(A P\) and \(B P\).
The other ends of the strings, \(A\) and \(B\), are attached to fixed points which are 4 metres apart on a rough horizontal surface at the bottom of a container. The coefficient of friction between \(P\) and the surface is 0.68

• When the extension of string \(A P\) is \(e _ { A }\) metres, the tension in \(A P\) is \(24 m e _ { A }\)
• When the extension of string \(B P\) is \(e _ { B }\) metres, the tension in \(B P\) is \(10 m e _ { B }\)
• The natural length of string \(A P\) is 1 metre
• The natural length of string \(B P\) is 1.3 metres
  \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-24_92_1082_1030_479}

15

1. Show that when \(A P = 1.5\) metres, the tension in \(A P\) is equal to the tension in \(B P\).
   15
2. \(\quad P\) is held at the point between \(A\) and \(B\) where \(A P = 1.9\) metres, and then released from rest. At time \(t\) seconds after \(P\) is released, \(A P = ( 1.5 + x )\) metres.
   \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-25_140_1068_493_484} Show that when \(P\) is moving towards \(A\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 34 x = 6.664$$ 15
3. The container is then filled with oil, and \(P\) is again released from rest at the point between \(A\) and \(B\) where \(A P = 1.9\) metres. At time \(t\) seconds after \(P\) is released, the oil causes a resistive force of magnitude \(10 m v\) newtons to act on the particle, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the particle. Find \(x\) in terms of \(t\) when \(P\) is moving towards \(A\).
   \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-27_2492_1721_217_150}
   \includegraphics[max width=\textwidth, alt={}]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-32_2486_1719_221_150}