6.02j Conservation with elastics: springs and strings

7 One end of a light elastic string of natural length 3 m and modulus of elasticity 24 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 64 + 20 x - 20 x ^ { 2 }\).
2. Find the greatest speed of the particle.
3. Calculate the greatest tension in the string.

3 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_247_839_1375_653} A light elastic string of natural length 1.2 m and modulus of elasticity 24 N is attached to fixed points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 1.2 \mathrm {~m}\). A particle \(P\) is attached to the mid-point of the string. \(P\) is projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the surface in a direction perpendicular to \(A B\) (see diagram). \(P\) comes to instantaneous rest at a distance 0.25 m from \(A B\).

1. Show that the mass of \(P\) is 0.8 kg .
2. Calculate the greatest deceleration of \(P\).

4 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 12 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.24 \mathrm {~kg} . P\) is projected vertically upwards with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a position 0.8 m vertically below \(O\).

1. Calculate the speed of the particle when it is moving upwards with zero acceleration.
2. Show that the particle moves 0.6 m while it is moving upwards with constant acceleration.

2 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 19.2 N . The other end of the string is attached to a fixed point \(A\). The particle \(P\) is released from rest at the point 2.7 m vertically above \(A\). Calculate

1. the initial acceleration of \(P\),
2. the speed of \(P\) when it reaches \(A\).

3 A particle \(P\) of mass 0.2 kg is attached to one end of a light elastic string of natural length 1.6 m and modulus of elasticity 18 N . The other end of the string is attached to a fixed point \(O\) which is 1.6 m above a smooth horizontal surface. \(P\) is placed on the surface vertically below \(O\) and then projected horizontally. \(P\) moves with initial speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on the surface. Show that, when \(O P = 1.8 \mathrm {~m}\),

1. \(P\) is at instantaneous rest,
2. \(P\) is on the point of losing contact with the surface.

5 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_341_529_260_808} A block \(B\) of mass 3 kg is attached to one end of a light elastic string of modulus of elasticity 70 N and natural length 1.4 m . The other end of the string is attached to a particle \(P\) of mass 0.3 kg . \(B\) is at rest 0.9 m from the edge of a horizontal table and the string passes over a small smooth pulley at the edge of the table. \(P\) is released from rest at a point next to the pulley and falls vertically. At the first instant when \(P\) is 0.8 m below the pulley and descending, \(B\) is in limiting equilibrium with the part of the string attached to \(B\) horizontal (see diagram).

1. Calculate the speed of \(P\) when \(B\) is first in limiting equilibrium.
2. Find the coefficient of friction between \(B\) and the table.

1 A light elastic string has modulus of elasticity 5 N and natural length 1.5 m . One end of the string is attached to a fixed point \(O\) and a particle \(P\) of mass 0.1 kg is attached to the other end of the string. \(P\) is released from rest at the point 2.4 m vertically below \(O\). Calculate the speed of \(P\) at the instant the string first becomes slack.

6 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-3_652_618_849_762} A particle \(P\) of mass 0.6 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 9 N . The string passes through a small smooth ring \(R\) fixed at a height of 0.4 m above a rough horizontal surface. The other end of the string is attached to a fixed point \(O\) which is 1.5 m vertically above \(R\). The points \(A\) and \(B\) are on the horizontal surface, and \(B\) is vertically below \(R\). When \(P\) is on the surface between \(A\) and \(B , R P\) makes an acute angle \(\theta ^ { \circ }\) with the horizontal (see diagram).

1. Show that the normal force exerted on \(P\) by the surface has magnitude 3.6 N , for all values of \(\theta\). \(P\) is projected with speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) from its initial position at \(A\) where \(\theta = 30\). The speed of \(P\) when it passes through \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
3. Calculate the value of the coefficient of friction between \(P\) and the surface.

2 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 30 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) which hangs in equilibrium vertically below \(O\), with \(O P = 0.8 \mathrm {~m}\).

1. Show that the mass of \(P\) is 1.8 kg . The particle is pulled vertically downwards and released from rest from the point where \(O P = 1.2 \mathrm {~m}\).
2. Find the speed of \(P\) at the instant when the string first becomes slack.

5 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the greatest speed of the particle during its descent.
2. Find the greatest distance of the particle below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-12_558_554_260_794} The end \(A\) of a non-uniform rod \(A B\) of length 0.6 m and weight 8 N rests on a rough horizontal plane, with \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude 3 N applied to the rod at \(B\). This force acts at \(30 ^ { \circ }\) above the horizontal in the vertical plane containing the rod (see diagram).
3. Find the distance of the centre of mass of the rod from \(A\).
   The 3 N force is removed, and the rod is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) applied at \(B\), acting in the vertical plane containing the rod, at an angle of \(30 ^ { \circ }\) below the horizontal.
4. Calculate \(P\).
   In one of the two situations described, the \(\operatorname { rod } A B\) is in limiting equilibrium.
5. Find the coefficient of friction at \(A\). \(7 \quad\) A particle \(P\) is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is \(y = 2 x - \frac { 25 x ^ { 2 } } { V ^ { 2 } }\).
6. Write down the value of \(\tan \theta\), where \(\theta\) is the angle of projection of \(P\).
   When \(t = 4 , P\) passes through the point \(A\) where \(x = y = a\).
7. Calculate \(V\) and \(a\).
8. Find the direction of motion of \(P\) when it passes through \(A\).

5 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point \(( 0.5 + x ) \mathrm { m }\) vertically below \(O\). The particle \(P\) comes to instantaneous rest at \(O\).

1. Find \(x\).
2. Find the greatest speed of \(P\).

7 A particle \(P\) of mass 0.5 kg is attached to a fixed point \(O\) by a light elastic string of natural length 1 m and modulus of elasticity 16 N . The particle \(P\) is projected vertically upwards from \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 x ^ { 2 } \mathrm {~N}\) acts on \(P\) when \(P\) has displacement \(x \mathrm {~m}\) above \(O\). After projection the upwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, before the string becomes taut, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.2 x ^ { 2 }\).
2. Find the velocity of \(P\) at the instant the string becomes taut.
3. Find an expression for the acceleration of \(P\) while it is moving upwards after the string becomes taut.
4. Verify that \(P\) comes to instantaneous rest before the extension of the string is 0.5 m .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 One end of a light elastic string of natural length 3 m and modulus of elasticity 15 mN is attached to a fixed point \(O\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. \(P\) is released from rest at \(O\) and moves vertically downwards. When the extension of the string is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 5 \left( 12 + 4 x - x ^ { 2 } \right)\).
2. Find the magnitude of the acceleration of \(P\) when it is at its lowest point, and state the direction of this acceleration.

6 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(k\), is attached to a particle \(P\) of mass \(m\). The other end of the string is attached to a fixed point \(Q\). The particle \(P\) is projected vertically upwards from \(Q\). When \(P\) is moving upwards and at a distance \(\frac { 4 } { 3 } a\) directly above \(Q\), it has a speed \(\sqrt { 2 g a }\). At this point, its acceleration is \(\frac { 7 } { 3 } g\) downwards. Show that \(\mathrm { k } = 4 \mathrm { mg }\) and find in terms of \(a\) the greatest height above \(Q\) reached by \(P\).

1 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The string hangs with \(P\) vertically below \(O\). The particle \(P\) is pulled vertically downwards so that the extension of the string is \(2 a\). The particle \(P\) is then released from rest.

1. Find the speed of \(P\) when it is at a distance \(\frac { 3 } { 4 } a\) below \(O\).
2. Find the initial acceleration of \(P\) when it is released from rest. \includegraphics[max width=\textwidth, alt={}, center]{454be64a-204f-4fa4-a5fc-72fd88e1289f-03_741_473_269_836} A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. It collides at an angle \(\alpha\) with a fixed smooth vertical barrier. After the collision, \(P\) moves at an angle \(\theta\) with the barrier, where \(\tan \theta = \frac { 1 } { 2 }\) (see diagram). The coefficient of restitution between \(P\) and the barrier is \(e\). The particle \(P\) loses 20\% of its kinetic energy as a result of the collision. Find the value of \(e\).

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]{454be64a-204f-4fa4-a5fc-72fd88e1289f-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 ( \(\mathrm { x } , \mathrm { y }\) ).

3 \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-2_397_1303_1790_422} The point \(O\) is 1.2 m below rough horizontal ground \(A B C\). A ball is projected from \(O\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) to the horizontal. The ball passes over the point \(A\) after travelling a horizontal distance of 2 m . The ball subsequently bounces once on the ground at \(B\). The ball leaves \(B\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and travels a further horizontal distance of 20 m before landing at \(C\) (see diagram).

1. Calculate the height above the level of \(O\) of the ball when it is vertically above \(A\).
2. Calculate the time after the instant of projection when the ball reaches \(B\).
3. Find the angle which the trajectory of the ball makes with the horizontal immediately after it bounces at \(B\). \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-3_663_695_258_726} A cylinder of height 0.9 m and radius 0.9 m is placed symmetrically on top of a cylinder of height \(h \mathrm {~m}\) and radius \(r \mathrm {~m}\), where \(r < 0.9\), with plane faces in contact and axes in the same vertical line \(A B\), where \(A\) and \(B\) are centres of plane faces of the cylinders (see diagram). Both cylinders are uniform and made of the same material. The lower cylinder is gradually tilted and when the axis of symmetry is inclined at \(45 ^ { \circ }\) to the horizontal the upper cylinder is on the point of toppling without sliding.

7 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 32 N . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).

1. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest. A horizontal plane is fixed at a distance 1 m below \(O\). The particle \(P\) is again released from rest at \(O\).
2. Calculate the speed of \(P\) immediately before it collides with the plane.
3. In the collision with the plane, \(P\) loses \(96 \%\) of its kinetic energy. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest above the plane, given that this occurs when the string is slack.

4. A light elastic string has natural length 0.4 m and modulus of elasticity 49 N . A particle \(P\) of mass 0.3 kg is attached to one end of the string. The other end of the string is attached to a fixed point \(A\) on a ceiling. The particle is released from rest at \(A\) and falls vertically. The particle first comes to instantaneous rest at the point \(B\).

1. Find the distance \(A B\). The particle is now held at the point 0.6 m vertically below \(A\) and released from rest.
2. Find the speed of \(P\) immediately before it hits the ceiling.

1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(2 m g\).

One end of the elastic string is attached to a fixed point \(O\). A particle \(P\) of mass \(\frac { 1 } { 2 } m\) is attached to the other end of the elastic string. The point \(A\) is vertically below \(O\) with \(O A = 4 a\). Particle \(P\) is held at \(A\) and released from rest. The speed of \(P\) at the instant when it has moved a distance \(a\) upwards is \(\sqrt { 3 a g }\) Air resistance to the motion of \(P\) is modelled as having magnitude \(k m g\), where \(k\) is a constant. Using the model and the work-energy principle,

1. show that \(k = \frac { 1 } { 4 }\) Particle \(P\) is now held at \(O\) and released from rest. As \(P\) moves downwards, it reaches its maximum speed as it passes through the point \(B\).
2. Find the distance \(O B\).

5. \begin{figure}[h] \end{figure} One end \(A\) of a light elastic string, of natural length \(a\) and modulus of elasticity \(6 m g\), is fixed at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. A small ball \(B\) of mass \(m\) is attached to the other end of the string. Initially \(B\) is held at rest with the string lying along a line of greatest slope of the plane, with \(B\) below \(A\) and \(A B = a\). The ball is released and comes to instantaneous rest at a point \(C\) on the plane, as shown in Figure 2. Find

1. the length \(A C\),
2. the greatest speed attained by \(B\) as it moves from its initial position to \(C\).

4. A rough disc rotates in a horizontal plane with constant angular velocity \(\omega\) about a fixed vertical axis. A particle \(P\) of mass \(m\) lies on the disc at a distance \(\frac { 4 } { 3 } a\) from the axis. The coefficient of friction between \(P\) and the disc is \(\frac { 3 } { 5 }\). Given that \(P\) remains at rest relative to the disc,

1. prove that \(\omega ^ { 2 } \leqslant \frac { 9 g } { 20 a }\). The particle is now connected to the axis by a horizontal light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The disc again rotates with constant angular velocity \(\omega\) about the axis and \(P\) remains at rest relative to the disc at a distance \(\frac { 4 } { 3 } a\) from the axis.
2. Find the greatest and least possible values of \(\omega ^ { 2 }\).

6. \begin{figure}[h] \end{figure} Figure 3 represents the path of a skier of mass 70 kg moving on a ski-slope \(A B C D\). The path lies in a vertical plane. From \(A\) to \(B\), the path is modelled as a straight line inclined at \(60 ^ { \circ }\) to the horizontal. From \(B\) to \(D\), the path is modelled as an arc of a vertical circle of radius 50 m . The lowest point of the \(\operatorname { arc } B D\) is \(C\). At \(B\), the skier is moving downwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At \(D\), the path is inclined at \(30 ^ { \circ }\) to the horizontal and the skier is moving upwards. By modelling the slope as smooth and the skier as a particle, find

1. the speed of the skier at \(C\),
2. the normal reaction of the slope on the skier at \(C\),
3. the speed of the skier at \(D\),
4. the change in the normal reaction of the slope on the skier as she passes \(B\). The model is refined to allow for the influence of friction on the motion of the skier.
5. State briefly, with a reason, how the answer to part (b) would be affected by using such a model. (No further calculations are expected.)

7. A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point \(O\) on a smooth horizontal table. The spring has natural length 2 m and modulus of elasticity 21.6 N . The particle \(P\) is placed on the table at the point \(A\), where \(O A = 2 \mathrm {~m}\). The particle \(P\) is now pulled away from \(O\) to the point \(B\), where \(O A B\) is a straight line with \(O B = 3.5 \mathrm {~m}\). It is then released from rest.

1. Prove that \(P\) moves with simple harmonic motion of period \(\frac { \pi } { 3 } \mathrm {~s}\).
2. Find the speed of \(P\) when it reaches \(A\). The point \(C\) is the mid-point of \(A B\).
3. Find, in terms of \(\pi\), the time taken for \(P\) to reach \(C\) for the first time. Later in the motion, \(P\) collides with a particle \(Q\) of mass 0.2 kg which is at rest at \(A\).
   After the impact, \(P\) and \(Q\) coalesce to form a single particle \(R\).
4. Show that \(R\) also moves with simple harmonic motion and find the amplitude of this motion. END

3. A light elastic string has natural length \(2 l\) and modulus of elasticity \(4 m g\). One end of the string is attached to a fixed point \(A\) and the other end to a fixed point \(B\), where \(A\) and \(B\) lie on a smooth horizontal table, with \(A B = 4 l\). A particle \(P\) of mass \(m\) is attached to the mid-point of the string. The particle is released from rest at the point of the line \(A B\) which is \(\frac { 5 l } { 3 }\) from \(B\). The speed of \(P\) at the mid-point of \(A B\) is \(V\).

1. Find \(V\) in terms of \(g\) and \(L\).
2. Explain why \(V\) is the maximum speed of \(P\).
   (Total 9 marks)