6.05d Variable speed circles: energy methods

4. \begin{figure}[h] \end{figure} A smooth hemisphere of radius \(a\) is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is \(O\). A particle \(P\) of mass \(M\) is disturbed from rest at the highest point of the hemisphere.
When \(P\) is still on the surface of the hemisphere and the radius from \(O\) to \(P\) is at an angle \(\theta\) to the vertical,

• the speed of \(P\) is \(v\)
• the normal reaction between the hemisphere and the particle is \(R\), as shown in Figure 2.
  1. Show that \(\mathrm { R } = \mathrm { Mg } ( 3 \cos \theta - 2 )\)
  2. Find, in terms of \(a\) and \(g\), the speed of the particle at the instant when the particle leaves the surface of the hemisphere.

7. \begin{figure}[h] \end{figure} A smooth solid hemisphere has radius \(r\) and the centre of its plane face is \(O\).
The hemisphere is fixed with its plane face in contact with horizontal ground, as shown in Figure 6.
A small stone is at the point \(A\), the highest point on the surface of the hemisphere. The stone is projected horizontally from \(A\) with speed \(U\).
The stone is still in contact with the hemisphere at the point \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical.
The speed of the stone at the instant it reaches \(B\) is \(v\).
The stone is modelled as a particle \(P\) and air resistance is modelled as being negligible.

1. Use the model to find \(v ^ { 2 }\) in terms of \(U , r , g\) and \(\theta\) When \(P\) leaves the surface of the hemisphere, the speed of \(P\) is \(W\).
   Given that \(U = \sqrt { \frac { 2 r g } { 3 } }\)
2. show that \(W ^ { 2 } = \frac { 8 } { 9 } r g\) After leaving the surface of the hemisphere, \(P\) moves freely under gravity until it hits the ground.
3. Find the speed of \(P\) as it hits the ground, giving your answer in terms of \(r\) and \(g\). At the instant when \(P\) hits the ground it is travelling at \(\alpha ^ { \circ }\) to the horizontal.
4. Find the value of \(\alpha\).

1. A small bead \(B\) of mass \(m\) is threaded on a circular hoop.

The hoop has centre \(O\) and radius \(a\) and is fixed in a vertical plane.
The bead is projected with speed \(\sqrt { \frac { 7 } { 2 } g a }\) from the lowest point of the hoop.
The hoop is modelled as being smooth.
When the angle between \(O B\) and the downward vertical is \(\theta\), the speed of \(B\) is \(v\).

1. Show that \(v ^ { 2 } = g a \left( \frac { 3 } { 2 } + 2 \cos \theta \right)\)
2. Find the size of \(\theta\) at the instant when the contact force between \(B\) and the hoop is first zero.
3. Give a reason why your answer to part (b) is not likely to be the actual value of \(\theta\).
4. Find the magnitude and direction of the acceleration of \(B\) at the instant when \(B\) is first at instantaneous rest.

3 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-2_473_298_1037_884} A particle \(P\) of mass 1.5 kg is attached to one end of a light inextensible string of length 2.4 m . The other end of the string is attached to a fixed point \(O\). The particle is initially at rest directly below \(O\). A horizontal impulse of magnitude 9.3 Ns is applied to \(P\). In the subsequent motion the string remains taut and makes an angle of \(\theta\) radians with the downwards vertical at \(O\), as shown in the diagram.

1. Find the speed of \(P\) when \(\theta = \frac { 1 } { 6 } \pi\).
2. Determine whether \(P\) will reach the same horizontal level as \(O\).

6 A fairground game involves a player kicking a ball, \(B\), from rest so as to project it with a horizontal velocity of magnitude \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is attached to one end of a light rod of length \(l \mathrm {~m}\). The other end of the rod is smoothly hinged at a fixed point \(O\) so that \(B\) can only move in the vertical plane which contains \(O\), a fixed barrier and a bell which is fixed \(l \mathrm {~m}\) vertically above \(O\). Initially \(B\) is vertically below \(O\). The barrier is positioned so that when \(B\) collides directly with the barrier, \(O B\) makes an angle \(\theta\) with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-4_643_659_584_724} The coefficient of restitution between \(B\) and the barrier is \(e . B\) rebounds from the barrier, passes through its original position and continues on a circular path towards the bell. The bell will only ring if the ball strikes it with a speed of at least \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The player wins the game if the player causes the bell to ring having kicked \(B\) so that it first collides with the barrier. You may assume that \(B\) and the bell are small and that the barrier has negligible thickness. Show that, whatever the position of the barrier, the player cannot win the game if \(u ^ { 2 } < 4 g l + \frac { V ^ { 2 } } { e ^ { 2 } }\). \section*{END OF QUESTION PAPER}

5 One end of a light inextensible string of length 0.8 m is attached to a fixed point, \(O\). The other end is attached to a particle \(P\) of mass \(1.2 \mathrm {~kg} . P\) hangs in equilibrium at a distance of 1.5 m above a horizontal plane. The point on the plane directly below \(O\) is \(F\). \(P\) is projected horizontally with speed \(3.5 \mathrm {~ms} ^ { - 1 }\). The string breaks when \(O P\) makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-3_776_910_1242_244}

1. Find the magnitude of the tension in the string at the instant before the string breaks.
2. Find the distance between \(F\) and the point where \(P\) first hits the plane.

2 \includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-02_810_743_831_644} A smooth wire is shaped into a circle of centre \(O\) and radius 0.8 m . The wire is fixed in a vertical plane. A small bead \(P\) of mass 0.03 kg is threaded on the wire and is projected along the wire from the highest point with a speed of \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(O P\) makes an angle \(\theta\) with the upward vertical the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).

1. Show that \(v ^ { 2 } = 33.32 - 15.68 \cos \theta\).
2. Prove that the bead is never at rest.
3. Find the maximum value of \(v\).
4. Write down the dimension of density. The workings of an oil pump consist of a right, solid cylinder which is partially submerged in oil. The cylinder is free to oscillate along its central axis which is vertical. If the base area of the pump is \(0.4 \mathrm {~m} ^ { 2 }\) and the density of the oil is \(920 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) then the period of oscillation of the pump is 0.7 s .
   A student assumes that the period of oscillation of the pump is dependent only on the density of the oil, \(\rho\), the acceleration due to gravity, \(g\), and the surface area, \(A\), of the circular base of the pump. The student attempts to test this assumption by stating that the period of oscillation, \(T\), is given by \(T = C \rho ^ { \alpha } g ^ { \beta } A ^ { \gamma }\) where \(C\) is a dimensionless constant.
5. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\).
6. Hence give the value of \(C\) to 3 significant figures.
7. Comment, with justification, on the assumption made by the student that the formula for the period of oscillation of the pump was dependent on only \(\rho , g\) and \(A\). A car of mass 1250 kg experiences a resistance to its motion of magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). At a point \(A\) on the road the car's speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it has an acceleration of magnitude \(0.54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At a point \(B\) on the road the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it has an acceleration of magnitude \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
8. Find the values of \(k\) and \(P\). The power is increased to 15 kW .
9. Calculate the maximum steady speed of the car on a straight horizontal road.

7 A hollow cylinder, of internal radius 4 m , is fixed so that its axis is horizontal. The point \(O\) is on this axis. A particle, of mass 6 kg , is set in motion so that it moves on the smooth inner surface of the cylinder in a vertical circle about \(O\). Its speed at the point \(A\), which is vertically below \(O\), is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-5_746_739_504_662} When the particle is at the point \(B\), at a height of 2 m above \(A\), find:

1. its speed;
2. the normal reaction between the cylinder and the particle.

7 A smooth hemisphere, of radius \(a\) and centre \(O\), is fixed with its plane face on a horizontal surface. A particle, of mass \(m\), can move freely on the surface of the hemisphere. The particle is placed at the point \(A\), the highest point of the hemisphere, and is set in motion along the surface with speed \(u\).

1. While the particle is in contact with the hemisphere at a point \(P , O P\) makes an angle \(\theta\) with the upward vertical. \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-5_366_1246_715_395} Show that the speed of the particle at \(P\) is $$\left( u ^ { 2 } + 2 g a [ 1 - \cos \theta ] \right) ^ { \frac { 1 } { 2 } }$$
2. The particle leaves the surface of the hemisphere when \(\theta = \alpha\). Find \(\cos \alpha\) in terms of \(a , u\) and \(g\).

7 A small bead, of mass \(m\), is suspended from a fixed point \(O\) by a light inextensible string, of length \(a\). The bead is then set into circular motion with the string taut at \(B\), where \(B\) is vertically below \(O\), with a horizontal speed \(u\). \includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-5_451_458_461_760}

1. Given that the string does not become slack, show that the least value of \(u\) required for the bead to make complete revolutions about \(O\) is \(\sqrt { 5 a g }\).
2. In the case where \(u = \sqrt { 5 a g }\), find, in terms of \(g\) and \(m\), the tension in the string when the bead is at the point \(C\), which is at the same horizontal level as \(O\), as shown in the diagram.
3. State one modelling assumption that you have made in your solution.

7 In crazy golf, a golf ball is hit so that it starts to move in a vertical circle on the inside of a smooth cylinder. Model the golf ball as a particle, \(P\), of mass \(m\). The circular path of the golf ball has radius \(a\) and centre \(O\). At time \(t\), the angle between \(O P\) and the horizontal is \(\theta\), as shown in the diagram. The golf ball has speed \(u\) at the lowest point of its circular path. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-6_739_742_719_641}

1. Show that, while the golf ball is in contact with the cylinder, the reaction of the cylinder on the golf ball is $$\frac { m u ^ { 2 } } { a } - 3 m g \sin \theta - 2 m g$$
2. Given that \(u = \sqrt { 3 a g }\), the golf ball will not complete a vertical circle inside the cylinder. Find the angle which \(O P\) makes with the horizontal when the golf ball leaves the surface of the cylinder.
   (4 marks)

7

1. Find Dominic's speed at the point when the cord initially becomes taut.
   7
2. Determine whether or not Dominic enters the river and gets wet.
   7
3. One limitation of this model is that Dominic is not a particle.
   Explain the effect of revising this assumption on your answer to part (b). \includegraphics[max width=\textwidth, alt={}, center]{1b79a789-c003-46c9-9235-254c1d8a0501-12_2492_1721_217_150} Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Additional page, if required.
   Write the question numbers in the left-hand margin.

1 \includegraphics[max width=\textwidth, alt={}, center]{d6a0d7a6-4166-4c26-a461-39b2414c0412-02_494_390_251_255} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(O B\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) (see diagram).

1. Find the speed of \(B\) at the instant when \(O B\) makes an angle of 0.8 radians with the downward vertical through \(O\).
2. Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\).

2 One end of a light inextensible string of length 0.75 m is attached to a particle \(A\) of mass 2.8 kg . The other end of the string is attached to a fixed point \(O . A\) is projected horizontally with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 0.75 m vertically above \(O\) (see Fig. 2). When \(O A\) makes an angle \(\theta\) with the upward vertical the speed of \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(\xrightarrow [ A \text { a } ] { 6 \mathrm {~m} \mathrm {~s} ^ { - 1 } }\) Fig. 2

1. Show that \(v ^ { 2 } = 50.7 - 14.7 \cos \theta\).
2. Given that the string breaks when the tension in it reaches 200 N , find the angle that \(O A\) turns through between the instant that \(A\) is projected and the instant that the string breaks.

2 One end of a light inextensible string of length 0.8 m is attached to a fixed point, \(O\). The other end is attached to a particle \(P\) of mass \(1.2 \mathrm {~kg} . P\) hangs in equilibrium at a distance of 1.5 m above a horizontal plane. The point on the plane directly below \(O\) is \(F\). \(P\) is projected horizontally with speed \(3.5 \mathrm {~ms} ^ { - 1 }\). The string breaks when \(O P\) makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{0428f2f2-12c4-4e89-93ab-8cfe2c5aca4a-02_757_889_1482_251}

1. Find the magnitude of the tension in the string at the instant before the string breaks.
2. Find the distance between \(F\) and the point where \(P\) first hits the plane.

A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt{\left(\frac{1}{2}ga\right)}\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(OP\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac{5}{2}mg(1 + 2\cos \theta)\). [4] Find the speed of \(P\)

1. when it loses contact with the sphere, [3]
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.) [3]

A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt{\left(\frac{1}{2}ga\right)}\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(OP\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac{5}{2}mg(1 + 2\cos\theta)\). [4] Find the speed of \(P\)

1. when it loses contact with the sphere, [3]
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.) [3]

\includegraphics{figure_5} A particle 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 point \(A\) is such that \(OA = a\) and \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is held at \(A\) and then projected downwards with speed \(\sqrt{(ag)}\) so that it begins to move in a vertical circle with centre \(O\). There is a small smooth peg at the point \(B\) which is at the same horizontal level as \(O\) and at a distance \(\frac{5}{4}a\) from \(O\) on the opposite side of \(O\) to \(A\) (see diagram).

1. Show that, when the string first makes contact with the peg, the speed of the particle is \(\sqrt{(ag(1 + 2\cos\alpha))}\). [2]

The particle now begins to move in a vertical circle with centre \(B\). When the particle is at the point \(C\) where angle \(CBO = 150°\), the tension in the string is the same as it was when the particle was at the point \(A\).

1. Find the value of \(\cos\alpha\). [10]

\includegraphics{figure_5} A particle 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 point \(A\) is such that \(OA = a\) and \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is held at \(A\) and then projected downwards with speed \(\sqrt{(ag)}\) so that it begins to move in a vertical circle with centre \(O\). There is a small smooth peg at the point \(B\) which is at the same horizontal level as \(O\) and at a distance \(\frac{3}{4}a\) from \(O\) on the opposite side of \(O\) to \(A\) (see diagram).

1. Show that, when the string first makes contact with the peg, the speed of the particle is \(\sqrt{(ag(1 + 2\cos\alpha))}\). [2]
2. The particle now begins to move in a vertical circle with centre \(B\). When the particle is at the point \(C\) where angle \(CBO = 150°\), the tension in the string is the same as it was when the particle was at the point \(A\). Find the value of \(\cos\alpha\). [10]

A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the lowest point on the inside of a fixed hollow sphere with centre \(O\). The sphere has a smooth internal surface of radius \(a\). Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to \(\frac{1}{2}u\) the angle \(\theta\) between \(OP\) and the downward vertical satisfies the equation $$8ga(1 - \cos\theta) = 3u^2.$$ [2] Find, in terms of \(m\), \(u\), \(a\) and \(g\), an expression for the magnitude of the contact force acting on the particle in this position. [4]

\includegraphics{figure_3} 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 \(AOB\) is a diameter of this circular cross-section and the radius \(OA\) 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 \(R_B = 10R_A\). Show that \(u^2 = ag\). [6] The particle loses contact with the surface of the cylinder when \(OP\) makes an angle \(\theta\) with the upward vertical. Find the value of \(\cos \theta\). [4]