6.02i Conservation of energy: mechanical energy principle

A rigid body consists of a thin uniform rod \(A B\), of mass \(4 m\) and length \(6 a\), joined at \(B\) to a point on the circumference of a uniform circular disc, with centre \(O\), mass \(8 m\) and radius \(2 a\). The point \(C\) on the circumference of the disc is such that \(B C\) is a diameter and \(A B C\) is a straight line (see diagram). The body rotates about a smooth fixed horizontal axis through \(C\), perpendicular to the plane of the disc. The angle between \(C A\) and the downward vertical at time \(t\) is denoted by \(\theta\).

1. Given that the body is performing small oscillations about the downward vertical, show that the period of these oscillations is approximately \(16 \pi \sqrt { } \left( \frac { a } { 11 g } \right)\).
2. Given instead that the body is released from rest in the position given by \(\cos \theta = 0.6\), find the maximum speed of \(A\).

3 \includegraphics[max width=\textwidth, alt={}, center]{2c6b6722-ebba-4ade-9a9d-dd70e61cf52b-2_413_414_1155_863} A smooth cylinder of radius \(a\) is fixed with its axis horizontal. The point \(O\) is the centre of a circular cross-section of the cylinder. The line \(A O B\) is a diameter of this circular cross-section and the radius \(O A\) makes an angle \(\alpha\) with the upward vertical (see diagram). It is given that \(\cos \alpha = \frac { 3 } { 5 }\). A particle \(P\) of mass \(m\) moves on the inner surface of the cylinder in the plane of the cross-section. The particle passes through \(A\) with speed \(u\) along the surface in the downwards direction. The magnitude of the reaction between \(P\) and the inner surface of the sphere is \(R _ { A }\) when \(P\) is at \(A\), and is \(R _ { B }\) when \(P\) is at \(B\). It is given that \(R _ { B } = 10 R _ { A }\). Show that \(u ^ { 2 } = a g\). The particle loses contact with the surface of the cylinder when \(O P\) makes an angle \(\theta\) with the upward vertical. Find the value of \(\cos \theta\).

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 \(P\).

3 \includegraphics[max width=\textwidth, alt={}, center]{5d40f5b4-e3d4-482c-8d8d-05a01bd3b43f-2_413_414_1155_863} A smooth cylinder of radius \(a\) is fixed with its axis horizontal. The point \(O\) is the centre of a circular cross-section of the cylinder. The line \(A O B\) is a diameter of this circular cross-section and the radius \(O A\) makes an angle \(\alpha\) with the upward vertical (see diagram). It is given that \(\cos \alpha = \frac { 3 } { 5 }\). A particle \(P\) of mass \(m\) moves on the inner surface of the cylinder in the plane of the cross-section. The particle passes through \(A\) with speed \(u\) along the surface in the downwards direction. The magnitude of the reaction between \(P\) and the inner surface of the sphere is \(R _ { A }\) when \(P\) is at \(A\), and is \(R _ { B }\) when \(P\) is at \(B\). It is given that \(R _ { B } = 10 R _ { A }\). Show that \(u ^ { 2 } = a g\). The particle loses contact with the surface of the cylinder when \(O P\) makes an angle \(\theta\) with the upward vertical. Find the value of \(\cos \theta\).

A uniform plane object consists of three identical circular rings, \(X , Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3 m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2 r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(A B\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left( 1 + \frac { 2 } { 3 } \sqrt { } 3 \right) r\).

1. Show that the moment of inertia of the object about \(A B\) is \(( 7 + 2 \sqrt { } 3 ) m r ^ { 2 }\). The line \(C D\) is the diameter of \(W\) that is perpendicular to \(A B\). A particle of mass \(9 m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(A B\).
2. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60 ^ { \circ }\).

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 \(O\). The particle is held vertically above \(O\) with the string taut and then projected horizontally with speed \(\sqrt { } \left( \frac { 13 } { 3 } a g \right)\). It begins to move in a vertical circle with centre \(O\). When \(P\) is at its lowest point, it collides with a stationary particle of mass \(\lambda m\). The two particles coalesce.

1. Show that the speed of the combined particle immediately after the impact is \(\frac { 5 } { \lambda + 1 } \sqrt { } \left( \frac { 1 } { 3 } a g \right)\). In the subsequent motion, the string becomes slack when the combined particle is at a height of \(\frac { 1 } { 3 } a\) above the level of \(O\).
2. Find the value of \(\lambda\).
3. Find, in terms of \(m\) and \(g\), the instantaneous change in the tension in the string as a result of the collision.

A thin uniform rod \(A B\) has mass \(2 m\) and length \(3 a\). Two identical uniform discs each have mass \(\frac { 1 } { 2 } m\) and radius \(a\). The centre of one of the discs is rigidly attached to the end \(A\) of the rod and the centre of the other disc is rigidly attached to the end \(B\) of the rod. The plane of each disc is perpendicular to the rod \(A B\). A second thin uniform rod \(O C\) has mass \(m\) and length \(2 a\). The end \(C\) of this rod is rigidly attached to the mid-point of \(A B\), with \(O C\) perpendicular to \(A B\) (see diagram). The object consisting of the two discs and two rods is free to rotate about a horizontal axis \(l\), through \(O\), which is perpendicular to both rods.

1. Show that the moment of inertia of one of the discs about \(l\) is \(\frac { 13 } { 4 } m a ^ { 2 }\).
2. Show that the moment of inertia of the object about \(l\) is \(\frac { 52 } { 3 } m a ^ { 2 }\). When the object is suspended from \(O\) and is hanging in equilibrium, the point \(C\) is given a speed of \(\sqrt { } ( 2 a g )\) in the direction parallel to \(A B\). In the subsequent motion, the angle through which \(O C\) has turned before the object comes to instantaneous rest is \(\theta\).
3. Show that \(\cos \theta = \frac { 8 } { 21 }\).

A thin uniform rod \(A B\) has mass \(2 m\) and length \(3 a\). Two identical uniform discs each have mass \(\frac { 1 } { 2 } m\) and radius \(a\). The centre of one of the discs is rigidly attached to the end \(A\) of the rod and the centre of the other disc is rigidly attached to the end \(B\) of the rod. The plane of each disc is perpendicular to the rod \(A B\). A second thin uniform rod \(O C\) has mass \(m\) and length \(2 a\). The end \(C\) of this rod is rigidly attached to the mid-point of \(A B\), with \(O C\) perpendicular to \(A B\) (see diagram). The object consisting of the two discs and two rods is free to rotate about a horizontal axis \(l\), through \(O\), which is perpendicular to both rods.

1. Show that the moment of inertia of one of the discs about \(l\) is \(\frac { 13 } { 4 } m a ^ { 2 }\).
2. Show that the moment of inertia of the object about \(l\) is \(\frac { 52 } { 3 } m a ^ { 2 }\). When the object is suspended from \(O\) and is hanging in equilibrium, the point \(C\) is given a speed of \(\sqrt { } ( 2 a g )\) in the direction parallel to \(A B\). In the subsequent motion, the angle through which \(O C\) has turned before the object comes to instantaneous rest is \(\theta\).
3. Show that \(\cos \theta = \frac { 8 } { 21 }\).

A thin uniform rod \(A B\) has mass \(2 m\) and length \(3 a\). Two identical uniform discs each have mass \(\frac { 1 } { 2 } m\) and radius \(a\). The centre of one of the discs is rigidly attached to the end \(A\) of the rod and the centre of the other disc is rigidly attached to the end \(B\) of the rod. The plane of each disc is perpendicular to the rod \(A B\). A second thin uniform rod \(O C\) has mass \(m\) and length \(2 a\). The end \(C\) of this rod is rigidly attached to the mid-point of \(A B\), with \(O C\) perpendicular to \(A B\) (see diagram). The object consisting of the two discs and two rods is free to rotate about a horizontal axis \(l\), through \(O\), which is perpendicular to both rods.

1. Show that the moment of inertia of one of the discs about \(l\) is \(\frac { 13 } { 4 } m a ^ { 2 }\).
2. Show that the moment of inertia of the object about \(l\) is \(\frac { 52 } { 3 } m a ^ { 2 }\). When the object is suspended from \(O\) and is hanging in equilibrium, the point \(C\) is given a speed of \(\sqrt { } ( 2 a g )\) in the direction parallel to \(A B\). In the subsequent motion, the angle through which \(O C\) has turned before the object comes to instantaneous rest is \(\theta\).
3. Show that \(\cos \theta = \frac { 8 } { 21 }\).

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).

2 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5 m\) and \(2 m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2 u\). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac { 1 } { 7 } u ( 1 + 15 e )\) and find an expression for the speed of \(A\).
   In the collision, the speed of \(A\) is halved and its direction of motion is reversed.
2. Find the value of \(e\).
3. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\). \includegraphics[max width=\textwidth, alt={}, center]{f2073c6e-0f76-4246-89a7-2f3a9f7aaff8-04_630_332_264_900} A uniform disc, of radius \(a\) and mass \(2 M\), is attached to a thin uniform rod \(A B\) of length \(6 a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).

1. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc.
   The object is free to rotate about the axis \(l\). The object is held with \(A B\) horizontal and is released from rest. When \(A B\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac { 3 } { 5 }\), the angular speed of the object is \(\sqrt { } \left( \frac { 2 g } { 5 a } \right)\).
2. Find the possible values of \(x\).

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 \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-10_1049_744_260_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).

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 \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-10_1051_744_258_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).

2 A small uniform sphere \(A\), of mass \(2 m\), is moving with speed \(u\) on a smooth horizontal surface when it collides directly with a small uniform sphere \(B\), of mass \(m\), which is at rest. The spheres have equal radii and the coefficient of restitution between them is \(e\).

1. Find expressions for the speeds of \(A\) and \(B\) immediately after the collision.
   Subsequently \(B\) collides with a vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is 0.4 . After \(B\) has collided with the wall, the speeds of \(A\) and \(B\) are equal.
2. Find \(e\).
3. Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\). \(3 A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(A B = 3 a \mathrm {~m}\). One end of a light elastic string, of natural length \(a\) m and modulus of elasticity \(m g \mathrm {~N}\), is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\). One end of a second light elastic string, of natural length \(k a \mathrm {~m}\) and modulus of elasticity \(2 m g \mathrm {~N}\), is attached to \(B\). The other end of this string is attached to \(P\). It is given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(A B\).

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.

2. A car of mass 800 kg is moving on a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving up the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(3 P\) watts. When the car is moving down the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(P\) watts.

1. Find
   1. the value of \(P\),
   2. the value of \(R\).
      (6) When the car is moving up the road at \(12.5 \mathrm {~ms} ^ { - 1 }\) the engine is switched off and the car comes to rest, without braking, in a distance \(d\) metres. The resistance to the motion of the car from non-gravitational forces is still modelled as a constant force of magnitude \(R\) newtons.
2. Use the work-energy principle to find the value of \(d\).

7 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-5_510_1091_269_479} \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal. \(M\) is the mid-point of \(A B\). Two particles \(P\) and \(Q\), joined by a taut light inextensible string, are placed on the plane at \(A\) and \(M\) respectively. The particles are simultaneously projected with speed \(0.6 \mathrm {~ms} ^ { - 1 }\) down the line of greatest slope (see diagram). The particles move down the plane with acceleration \(0.9 \mathrm {~ms} ^ { - 2 }\). At the instant 2 s after projection, \(P\) is at \(M\) and \(Q\) is at \(B\). The particle \(Q\) subsequently remains at rest at \(B\).

1. Find the distance \(A B\). The plane is rough between \(A\) and \(M\), but smooth between \(M\) and \(B\).
2. Calculate the speed of \(P\) when it reaches \(B\). \(P\) has mass 0.4 kg and \(Q\) has mass 0.3 kg .
3. By considering the motion of \(Q\), calculate the tension in the string while both particles are moving down the plane.
4. Calculate the coefficient of friction between \(P\) and the plane between \(A\) and \(M\). \section*{END OF QUESTION PAPER}

6 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg . There is a constant resistance force of magnitude 32 N to the cyclist's motion. At an instant when she is travelling at \(7 \mathrm {~ms} ^ { - 1 }\), her acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).

1. Find the power output of the cyclist.
2. Find the steady speed that the cyclist can maintain if her power output and the resistance force are both unchanged. \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-08_2718_35_141_2012} \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-09_2724_35_136_20} The cyclist later descends a straight hill of length 32.2 m , inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 20 } \right)\) to the horizontal. Her power output is now 120 W , and the resistance force now has variable magnitude such that the work done against this force in descending the hill is 1128 J . The time taken to descend the hill is 4 s .
3. Given that the speed of the cyclist at the top of the hill is \(7.5 \mathrm {~ms} ^ { - 1 }\), find her speed at the bottom of the hill.

7 \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_323_1308_292_376} The diagram shows a track \(A B C D\) which lies in a vertical plane. The section \(A B\) is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is smooth. The section \(B C\) is a horizontal straight line and is rough. The section CD is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is rough. The lengths \(A B , B C\) and \(C D\) are each 2 m . A particle is released from rest at \(A\). The coefficient of friction between the particle and both \(B C\) and \(C D\) is \(\mu\). There is no change in the speed of the particle when it passes through either of the points \(B\) or \(C\).

1. It is given that \(\mu = 0.1\). Find the distance which the particle has moved up the section \(C D\) when its speed is \(1 \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_2716_33_143_2014}
2. It is given instead that with a different value of \(\mu\) the particle travels 1 m up the track from \(C\) before it comes instantaneously to rest. Find the value of \(\mu\) and the speed of the particle at the instant that it passes \(C\) for the second time.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 A toy boat of mass 1.5 kg is pushed across a pond, starting from rest, for 2.5 seconds. During this time, the boat has an acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Subsequently, when the only horizontal force acting on the boat is a constant resistance to motion, the boat travels 10 m before coming to rest. Calculate the magnitude of the resistance to motion.

9 A child throws a pebble of mass 40 g vertically downwards with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) from a point 0.8 m above a sandy beach.

1. Calculate the speed at which the pebble hits the beach. The pebble travels 3 cm through the sand before coming to rest.
2. Find the magnitude of the resistance force of the sand on the pebble, assuming it is constant. Give your answer correct to \(\mathbf { 3 }\) significant figures.

7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) As part of a competition, Jo-Jo makes a small pop-up rocket.
It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
The mass of the rocket is 18 grams and the stiffness constant of the spring is \(60 \mathrm { Nm } ^ { - 1 }\) Initially the spring is compressed by 3 cm
7

1. Find the speed of the rocket when the spring first reaches its natural length.
   7
2. By considering energy find the distance that the rocket rises. 7
3. In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.

4 A cricket ball of mass 156 grams is thrown from a point which is 1.5 metres above the ground, with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) A tennis ball of mass 58 grams is thrown from the same point, with the same speed.
Prove that both balls hit the ground with the same speed.
Clearly state any assumptions you have made and how you have used them.
[0pt] [5 marks]