Particle on smooth curved surface

3 \includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-04_519_1018_260_561} The diagram shows the vertical cross-section of a surface. \(A , B\) and \(C\) are three points on the crosssection. The level of \(B\) is \(h \mathrm {~m}\) above the level of \(A\). The level of \(C\) is 0.5 m below the level of \(A\). A particle of mass 0.2 kg is projected up the slope from \(A\) with initial speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle remains in contact with the surface as it travels from \(A\) to \(C\).

1. Given that the particle reaches \(B\) with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there is no resistance force, find \(h\).
2. It is given instead that there is a resistance force and that the particle does 3.1 J of work against the resistance force as it travels from \(A\) to \(C\). Find the speed of the particle when it reaches \(C\).

3 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-04_416_792_260_674} The diagram shows a semi-circular track \(A B C\) of radius 1.8 m which is fixed in a vertical plane. The points \(A\) and \(C\) are at the same horizontal level and the point \(B\) is at the bottom of the track. The section \(A B\) is smooth and the section \(B C\) is rough. A small block is released from rest at \(A\).

1. Show that the speed of the block at \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The block comes to instantaneous rest for the first time at a height of 1.2 m above the level of \(B\). The work done against the resistance force during the motion of the block from \(B\) to this point is 4.5 J .
2. Find the mass of the block.

4 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_478_1041_269_552} \(O A B C\) is a vertical cross-section of a smooth surface. The straight part \(O A\) has length 2.4 m and makes an angle of \(50 ^ { \circ }\) with the horizontal. \(A\) and \(C\) are at the same horizontal level and \(B\) is the lowest point of the cross-section (see diagram). A particle \(P\) of mass 0.8 kg is released from rest at \(O\) and moves on the surface. \(P\) remains in contact with the surface until it leaves the surface at \(C\). Find

1. the kinetic energy of \(P\) at \(A\),
2. the speed of \(P\) at \(C\). The greatest speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the depth of \(B\) below the horizontal through \(A\) and \(C\).

4 \includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-2_227_586_1631_781} The diagram shows a vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section. A particle of mass 0.15 kg is released from rest at \(A\).

1. Assuming that the particle reaches \(B\) with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there are no resistances to motion, find the height of \(A\) above \(B\).
2. Assuming instead that the particle reaches \(B\) with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) and that the height of \(A\) above \(B\) is 4 m , find the work done against the resistances to motion.

5 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-3_515_789_995_676} The diagram shows the vertical cross-section \(L M N\) of a fixed smooth surface. \(M\) is the lowest point of the cross-section. \(L\) is 2.45 m above the level of \(M\), and \(N\) is 1.2 m above the level of \(M\). A particle of mass 0.5 kg is released from rest at \(L\) and moves on the surface until it leaves it at \(N\). Find

1. the greatest speed of the particle,
2. the kinetic energy of the particle at \(N\). The particle is now projected from \(N\), with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface towards \(M\).
3. Find the least value of \(v\) for which the particle will reach \(L\).

2 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-2_609_967_536_589} A smooth narrow tube \(A E\) has two straight parts, \(A B\) and \(D E\), and a curved part \(B C D\). The part \(A B\) is vertical with \(A\) above \(B\), and \(D E\) is horizontal. \(C\) is the lowest point of the tube and is 0.65 m below the level of \(D E\). A particle is released from rest at \(A\) and travels through the tube, leaving it at \(E\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) (see diagram). Find

1. the height of \(A\) above the level of \(D E\),
2. the maximum speed of the particle.

6 \includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-3_382_1451_797_347} The diagram shows the vertical cross-section \(A B C D\) of a surface. \(B C\) is a circular arc, and \(A B\) and \(C D\) are tangents to \(B C\) at \(B\) and \(C\) respectively. \(A\) and \(D\) are at the same horizontal level, and \(B\) and \(C\) are at heights 2.7 m and 3.0 m respectively above the level of \(A\) and \(D\). A particle \(P\) of mass 0.2 kg is given a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\), in the direction of \(A B\) (see diagram). The parts of the surface containing \(A B\) and \(B C\) are smooth.

1. Find the decrease in the speed of \(P\) as \(P\) moves along the surface from \(B\) to \(C\). The part of the surface containing \(C D\) exerts a constant frictional force on \(P\), as it moves from \(C\) to \(D\), and \(P\) comes to rest as it reaches \(D\).
2. Find the speed of \(P\) when it is at the mid-point of \(C D\).

1 \includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-2_366_780_251_680} \(A B C D\) is a semi-circular cross-section, in a vertical plane, of the inner surface of half a hollow cylinder of radius 2.5 m which is fixed with its axis horizontal. \(A D\) is horizontal, \(B\) is the lowest point of the cross-section and \(C\) is at a height of 1.8 m above the level of \(B\) (see diagram). A particle \(P\) of mass 0.8 kg is released from rest at \(A\) and comes to instantaneous rest at \(C\).

1. Find the work done on \(P\) by the resistance to motion while \(P\) travels from \(A\) to \(C\). The work done on \(P\) by the resistance to motion while \(P\) travels from \(A\) to \(B\) is 0.6 times the work done while \(P\) travels from \(A\) to \(C\).
2. Find the speed of \(P\) when it passes through \(B\).

2 \includegraphics[max width=\textwidth, alt={}, center]{b190b8c9-75b0-4ede-913f-cdecdb58180f-2_337_579_842_246} A small body \(P\) of mass 3 kg is at rest at the lowest point of the inside of a smooth hemispherical shell of radius 3.2 m and centre \(O\). \(P\) is projected horizontally with a speed of \(u \mathrm {~ms} ^ { - 1 }\). When \(P\) first comes to instantaneous rest \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical through \(O\).

1. Find the value of \(u\).
2. State one assumption made in modelling the motion of \(P\).

5 A bead of mass \(m\) moves on a smooth circular ring of radius \(a\) which is fixed in a vertical plane, as shown in the diagram. Its speed at \(A\), the highest point of its path, is \(v\) and its speed at \(B\), the lowest point of its path, is \(7 v\). \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-4_419_317_484_842}

1. Show that \(v = \sqrt { \frac { a g } { 12 } }\).
2. Find the reaction of the ring on the bead, in terms of \(m\) and \(g\), when the bead is at \(A\).

6 Simon, a small child of mass 22 kg , is on a swing. He is swinging freely through an angle of \(18 ^ { \circ }\) on both sides of the vertical. Model Simon as a particle, \(P\), of mass 22 kg , attached to a fixed point, \(Q\), by a light inextensible rope of length 2.4 m . \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-5_700_310_466_849}

1. Find Simon's maximum speed as he swings.
2. Calculate the tension in the rope when Simon's speed is a maximum.

6 A particle \(P\), of mass \(m \mathrm {~kg}\), is placed at the point \(Q\) on the top of a smooth upturned hemisphere of radius 3 metres and centre \(O\). The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of \(2 \mathrm {~ms} ^ { - 1 }\). When the particle is on the surface of the hemisphere, the angle between \(O P\) and \(O Q\) is \(\theta\) and the particle has speed \(v \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-005_419_1013_607_511}

1. Show that \(v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )\).
2. Find the value of \(\theta\) when the particle leaves the hemisphere.

4 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) An inelastic string has length 1.2 metres.
One end of the string is attached to a fixed point \(O\).
A sphere, of mass 500 grams, is attached to the other end of the string.
The sphere is held, with the string taut and at an angle of \(20 ^ { \circ }\) to the vertical, touching the chin of a student, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-04_739_511_676_762} The sphere is released from rest.
Assume that the student stays perfectly still once the sphere has been released.
4

1. Calculate the maximum speed of the sphere.
   

   4

A child of mass \(35\text{ kg}\) is swinging on a rope. The child is modelled as a particle \(P\) and the rope is modelled as a light inextensible string of length \(4\text{ m}\). Initially \(P\) is held at an angle of \(45°\) to the vertical (see diagram). \includegraphics{figure_5}

1. Given that there is no resistance force, find the speed of \(P\) when it has travelled half way along the circular arc from its initial position to its lowest point. [4]
2. It is given instead that there is a resistance force. The work done against the resistance force as \(P\) travels from its initial position to its lowest point is \(X\text{ J}\). The speed of \(P\) at its lowest point is \(4\text{ m s}^{-1}\). Find \(X\). [3]

A fixed smooth sphere has centre \(O\) and radius \(a\). A particle \(P\) is placed on the surface of the sphere at the point \(A\), where \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is released from rest at \(A\). When \(OP\) makes an angle \(\theta\) to the upward vertical through \(O\), \(P\) is on the surface of the sphere and the speed of \(P\) is \(v\). Given that \(\cos \alpha = \frac{3}{5}\)

1. show that $$v^2 = \frac{2ga}{5}(3 - 5\cos \theta)$$ [4]
2. find the speed of \(P\) at the instant when it loses contact with the sphere. [8]

\includegraphics{figure_2} A smooth sphere of radius \(a\) is fixed with a point \(A\) of its surface in contact with a fixed vertical wall. A particle is placed on the highest point of the sphere and is projected towards the wall and perpendicular to the wall with horizontal speed \(\sqrt{\frac{2ag}{5}}\), as shown in Figure 2. The particle leaves the surface of the sphere with speed \(V\).

1. Show that \(V = \sqrt{\frac{4ag}{5}}\) [7]

The particle strikes the wall at the point \(X\).

1. Find the distance \(AX\). [9]

A light spring, of natural length 30 cm, is fixed in a vertical position. When a small ball of mass 0.4 kg rests on top of it, the spring is compressed by 10 cm. The ball is then held at a height of 15 cm vertically above the top of the spring and released from rest. Calculate the maximum compression of the string in the resulting motion. [7 marks]

A particle of mass \(m\) kg is attached to one end of a light inextensible string of length \(l\) m whose other end is fixed to a point \(O\). The particle is made to move in a vertical circle with centre \(O\), with constant angular velocity \(\omega\) rad s\(^{-1}\). At a certain instant it is in the position shown, where the string makes an angle \(\theta\) radians with the downward vertical through \(O\). \includegraphics{figure_1}

1. Find an expression, in terms of \(m\), \(l\) and \(\omega\), for the kinetic energy of the particle at this instant. [2 marks]
2. Find an expression, in terms of \(m\), \(g\), \(l\) and \(\theta\), for the potential energy of the particle relative to the horizontal plane through the lowest point \(A\). [2 marks]
3. Determine the position of the particle when the rate of increase of its total energy, with respect to time, is a maximum. [3 marks]

\includegraphics{figure_7} 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 \(OB\) 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 \(OB\) makes an angle of 0.8 radians with the downward vertical through \(O\). [3]
2. Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\). [3]

A particle \(P\) of mass 5.6 kg is attached to one end of a light rod of length 2.1 m. The other end of the rod is freely hinged to a fixed point \(O\). The particle is initially at rest directly below \(O\). It is then projected horizontally with speed \(5 \text{ ms}^{-1}\). In the subsequent motion, the angle between the rod and the downward vertical at \(O\) is denoted by \(\theta\) radians, as shown in the diagram. \includegraphics{figure_2}

1. Find the speed of \(P\) when \(\theta = \frac{1}{4}\pi\). [4]
2. Find the value of \(\theta\) when \(P\) first comes to instantaneous rest. [2]