SHM on inclined plane

7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The string lies along a line of greatest slope of the plane. The particle rests in equilibrium at the point \(B\), where \(B\) is lower than \(A\) and \(A B = \frac { 6 } { 5 } a\).

1. Show that \(\lambda = \frac { 5 } { 2 } m g\). The particle is now pulled down a line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 5 } a\), and released from rest.
2. Show that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \frac { 2 a } { 5 g } }\)
3. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(P\) while the string is taut. The midpoint of \(B C\) is \(D\) and the string becomes slack for the first time at the point \(E\).
4. Find, in terms of \(a\) and \(g\), the time taken by \(P\) to travel directly from \(D\) to \(E\).

7. \begin{figure}[h] \end{figure} A small ring \(R\) of mass in is free to slide on a smooth straight wire which is fixed at an angle of \(30 ^ { \circ }\) to the horizontal. The ring is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point \(B\), where \(A B = \frac { 9 } { 8 } a\).

1. Show that \(\lambda = 4 m g\). The ring is pulled down to the point \(C\), where \(B C = \frac { 1 } { 4 } a\), and released from rest. At time \(t\) after \(R\) is released the extension of the string is ( \(\frac { 1 } { 8 } a + x\) ).
2. Obtain a differential equation for the motion of \(R\) while the string remains taut, and show that it represents simple harmonic motion with period \(\pi \sqrt { \left( \frac { a } { g } \right) }\).
   (6)
3. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(R\) while the string remains taut.
   (2)
4. Find, in terms of \(a\) and \(g\), the time taken for \(R\) to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. END

A particle \(P\) of mass \(3 m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(k m g\). The other end of the spring is attached to a fixed point \(O\) on a smooth plane that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 2 } { 3 }\). The system rests in equilibrium with \(P\) on the plane at the point \(E\). The length of the spring in this position is \(\frac { 5 } { 4 } a\).

1. Find the value of \(k\).
   The particle \(P\) is now replaced by a particle \(Q\) of mass \(2 m\) and \(Q\) is released from rest at the point \(E\).
2. Show that, in the resulting motion, \(Q\) performs simple harmonic motion. State the centre and the period of the motion.
3. Find the least tension in the spring and the maximum acceleration of \(Q\) during the motion.

7 A particle \(P\) of mass 0.5 kg is attached to one end of each of two identical light elastic strings of natural length 1.6 m and modulus of elasticity 19.6 N . The other ends of the strings are attached to fixed points \(A\) and \(B\) on a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is 4.8 m and \(A\) is higher than \(B\).

1. Find the distance \(A P\) for which \(P\) is in equilibrium on the line \(A B\).
   \(P\) is released from rest at a point on \(A B\) where both strings are taut. The strings remain taut during the subsequent motion of \(P\) and \(t\) seconds after release the distance \(A P\) is \(( 2.5 + x ) \mathrm { m }\).
2. Use Newton's second law to obtain an equation of the form \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = k x\). State the property of the constant \(k\) for which the equation indicates that \(P\) 's motion is simple harmonic, and find the period of this motion.
3. Given that \(x = 0.5\) when \(t = 0\), find the values of \(x\) for which the speed of \(P\) is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). {www.ocr.org.uk}) after the live examination series.
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7
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_382_773_1567_648} One end of a light elastic string, of natural length 0.3 m , is attached to a fixed point \(O\) on a smooth plane that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.2\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. The string lies along a line of greatest slope of the plane and has modulus of elasticity \(2.45 m \mathrm {~N}\) (see diagram).

1. Show that in the equilibrium position the extension of the string is 0.24 m .
   \(P\) is given a velocity of \(0.3 \mathrm {~ms} ^ { - 1 }\) down the plane from the equilibrium position.
2. Show that \(P\) performs simple harmonic motion with period 2.20 s (correct to 3 significant figures), and find the amplitude of the motion.
3. Find the distance of \(P\) from \(O\) and the velocity of \(P\) at the instant 1.5 seconds after \(P\) is set in motion.

4 Fig. 4 shows a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Two fixed points A and B on the plane are 4.55 m apart with B higher than A on a line of greatest slope. A particle P of mass 0.25 kg is in contact with the plane and is connected to A and to B by two light elastic strings. The string AP has natural length 1.5 m and modulus of elasticity 7.35 N ; the string BP has natural length 2.5 m and modulus of elasticity 7.35 N . The particle P moves along part of the line AB , with both strings taut throughout the motion. \begin{figure}[h] \end{figure}

1. Show that, when \(\mathrm { AP } = 1.55 \mathrm {~m}\), the acceleration of P is zero.
2. Taking \(\mathrm { AP } = ( 1.55 + x ) \mathrm { m }\), write down the tension in the string AP , in terms of \(x\), and show that the tension in the string BP is \(( 1.47 - 2.94 x ) \mathrm { N }\).
3. Show that the motion of P is simple harmonic, and find its period. The particle P is released from rest with \(\mathrm { AP } = 1.5 \mathrm {~m}\).
4. Find the time after release when P is first moving down the plane with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

11 In this question use \(g\) as \(10 \mathrm {~m \mathrm {~s} ^ { - 2 }\)} A smooth plane is inclined at \(30 ^ { \circ }\) to the horizontal.
The fixed points \(A\) and \(B\) are 3.6 metres apart on the line of greatest slope of the plane, with \(A\) higher than \(B\) A particle \(P\) of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points \(A\) and \(B\) respectively. The particle \(P\) moves on a straight line that passes through \(A\) and \(B\)
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-18_417_709_774_669} The natural length of the string \(A P\) is 1.4 metres.
When the extension of the string \(A P\) is \(e _ { A }\) metres, the tension in the string \(A P\) is \(7 e _ { A }\) newtons.
The natural length of the string \(B P\) is 1 metre.
When the extension of the string \(B P\) is \(e _ { B }\) metres, the tension in the string \(B P\) is \(9 e _ { B }\) newtons. The particle \(P\) is held at the point between \(A\) and \(B\) which is 0.2 metres from its equilibrium position and lower than its equilibrium position.
The particle \(P\) is then released from rest.
At time \(t\) seconds after \(P\) is released, its displacement towards \(B\) from its equilibrium position is \(x\) metres. 11

1. Show that during the subsequent motion the object satisfies the equation $$\ddot { x } + 50 x = 0$$ Fully justify your answer. 11
2. The experiment is repeated in a large tank of oil.
   During the motion the oil causes a resistive force of \(k v\) newtons to act on the particle, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the particle. The oil causes critical damping to occur.
   11
3. 1. Show that \(k = \frac { 16 \sqrt { 2 } } { 5 }\)
      11
4. (ii) Find \(x\) in terms of \(t\), giving your answer in exact form.
   11
5. (iii) Calculate the maximum speed of the particle.