Particle on outer surface of sphere

6. \begin{figure}[h] \end{figure} A smooth solid hemisphere of radius 0.5 m is fixed with its plane face on a horizontal floor. The plane face has centre \(O\) and the highest point of the surface of the hemisphere is \(A\). A particle \(P\) has mass 0.2 kg . The particle is projected horizontally with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) and leaves the hemisphere at the point \(B\), where \(O B\) makes an angle \(\theta\) with \(O A\), as shown in Figure 1. The point \(B\) is at a vertical distance of 0.1 m below the level of \(A\). The speed of \(P\) at \(B\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Show that \(v ^ { 2 } = u ^ { 2 } + 1.96\)
2. Find the value of \(u\). The particle first strikes the floor at the point \(C\).
3. Find the length of \(O C\).

7. \begin{figure}[h] \end{figure} A smooth solid sphere, with centre \(O\) and radius \(r\), is fixed with its lowest point on a horizontal plane. A particle is placed on the surface of the sphere at the highest point of the sphere. The particle is then projected horizontally with speed \(u\) and starts to move on the surface of the sphere. The particle leaves the surface of the sphere at the point \(A\) where \(O A\) makes an angle \(\alpha , \alpha > 0\), with the upward vertical, as shown in Figure 4.

1. Show that \(\cos \alpha = \frac { 1 } { 3 g r } \left( u ^ { 2 } + 2 g r \right)\)
2. Show that \(u < \sqrt { g r }\) After leaving the surface of the sphere, the particle strikes the plane with speed \(3 \sqrt { \frac { g r } { 2 } }\)
3. Find the value of \(\cos \alpha\).

6. \begin{figure}[h] \end{figure} A fixed solid sphere has centre \(O\) and radius \(r\).
A particle \(P\) of mass \(m\) is held at rest on the smooth surface of the sphere at \(A\), the highest point of the sphere.
The particle \(P\) is then projected horizontally from \(A\) with speed \(u\) and moves on the surface of the sphere.
At the instant when \(P\) reaches the point \(B\) on the sphere, where angle \(A O B = \theta , P\) is moving with speed \(v\), as shown in Figure 4. At this instant, \(P\) loses contact with the surface of the sphere.

1. Show that $$\cos \theta = \frac { 2 g r + u ^ { 2 } } { 3 g r }$$ In the subsequent motion, the particle \(P\) crosses the horizontal through \(O\) at the point \(C\), also shown in Figure 4. At the instant \(P\) passes through \(C , P\) is moving at an angle \(\alpha\) to the horizontal.
   Given that \(u ^ { 2 } = \frac { 2 g r } { 5 }\)
2. find the exact value of \(\tan \alpha\).

7. \begin{figure}[h] \end{figure} A particle is projected from the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g a )\). The particle leaves the surface of the sphere at the point \(C\), where \(\angle A O C = \theta\), and strikes the plane at the point \(P\), as shown in Figure 5.

1. Show that \(\cos \theta = \frac { 3 } { 4 }\).
2. Find the angle that the velocity of the particle makes with the horizontal as it reaches \(P\).

6. \begin{figure}[h] \end{figure} A particle is at the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(u\), where \(u < \sqrt { } ( a g )\). The particle leaves the sphere at the point \(C\), where \(O C\) makes an angle \(\theta\) with the upward vertical, as shown in Fig. 2.

1. Find an expression for \(\cos \theta\) in terms of \(u , g\) and \(a\). The particle strikes the plane with speed \(\sqrt { \left( \frac { 9 a g } { 2 } \right) }\).
2. Find, to the nearest degree, the value of \(\theta\).

5. A smooth solid sphere, with centre \(O\) and radius \(a\), is fixed to the upper surface of a horizontal table. A particle \(P\) is placed on the surface of the sphere at a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical, and \(0 < \alpha < \frac { \pi } { 2 }\). The particle is released from rest. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still on the surface of the sphere, the speed of \(P\) is \(v\).

1. Show that \(v ^ { 2 } = 2 g a ( \cos \alpha - \cos \theta )\). Given that \(\cos \alpha = \frac { 3 } { 4 }\), find
2. the value of \(\theta\) when \(P\) loses contact with the sphere,
3. the speed of \(P\) as it hits the table.
   (Total 13 marks)

1. 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 \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is released from rest at \(A\). When \(O P\) 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 $$v ^ { 2 } = \frac { 2 g a } { 5 } ( 3 - 5 \cos \theta )$$
2. find the speed of \(P\) at the instant when it loses contact with the sphere.

4. \begin{figure}[h] \end{figure} 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 \(\sqrt { \frac { 2 a g } { 5 } }\), as shown in Figure 2. The particle leaves the surface of the sphere with speed \(V\).

1. Show that \(V = \sqrt { \frac { 4 a g } { 5 } }\) The particle strikes the wall at the point \(X\).
2. Find the distance \(A X\).

5 A particle \(P\) of mass \(m\) is placed at the point \(Q\) on the outer surface of a fixed smooth sphere with centre \(O\) and radius \(a\). The acute angle between \(O Q\) and the upward vertical is \(\alpha\), where \(\cos \alpha = \frac { 9 } { 10 }\). The particle is released from rest and begins to move in a vertical circle on the surface of the sphere. Show that \(P\) loses contact with the sphere when \(O P\) makes an angle \(\theta\) with the upward vertical, where \(\cos \theta = \frac { 3 } { 5 }\), and find the speed of \(P\) at this instant. Show that, in the subsequent motion, when \(P\) is at a distance \(\frac { 7 } { 5 } a\) from the vertical diameter through \(O\), its distance below the horizontal through \(O\) is \(\frac { 31 } { 30 } a\).

A smooth sphere, with centre \(O\) and radius \(a\), is fixed on a smooth horizontal plane \(\Pi\). A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(\sqrt { } \left( \frac { 2 } { 5 } g a \right)\). While \(P\) remains in contact with the sphere, the angle between \(O P\) and the upward vertical is denoted by \(\theta\). Show that \(P\) loses contact with the sphere when \(\cos \theta = \frac { 4 } { 5 }\). Subsequently the particle collides with the plane \(\Pi\). The coefficient of restitution between \(P\) and \(\Pi\) is \(\frac { 5 } { 9 }\). Find the vertical height of \(P\) above \(\Pi\) when the vertical component of the velocity of \(P\) first becomes zero.

7
\includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-4_330_1061_989_267} The flat surface of a smooth solid hemisphere of radius \(r\) is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by \(\gamma\). \(O\) is the centre of the flat surface of the hemisphere. A particle \(P\) is held at a point on the surface of the hemisphere such that the angle between \(O P\) and the upward vertical through \(O\) is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 4 }\).
\(P\) is then released from rest. \(F\) is the point on the plane where \(P\) first hits the plane (see diagram).

1. Find an exact expression for the distance \(O F\). The acceleration due to gravity on and near the surface of the planet Earth is roughly \(6 \gamma\).
2. Explain whether \(O F\) would increase, decrease or remain unchanged if the action were repeated on the planet Earth. \section*{END OF QUESTION PAPER}

7
\includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-4_517_677_267_733} A particle \(P\) of mass \(m \mathrm {~kg}\) is slightly disturbed from rest at the highest point on the surface of a smooth fixed sphere of radius \(a\) m and centre \(O\). The particle starts to move downwards on the surface. While \(P\) remains on the surface \(O P\) makes an angle of \(\theta\) radians with the upward vertical and has angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram). The sphere exerts a force of magnitude \(R \mathrm {~N}\) on \(P\).

1. Show that \(a \omega ^ { 2 } = 2 g ( 1 - \cos \theta )\).
2. Find an expression for \(R\) in terms of \(m , g\) and \(\theta\). At the instant that \(P\) loses contact with the surface of the sphere, find
3. the transverse component of the acceleration of \(P\),
4. the rate of change of \(R\) with respect to time \(t\), in terms of \(m , g\) and \(a\).

5
\includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_449_442_1281_794} A fixed smooth sphere of radius 0.6 m has centre \(O\) and highest point \(T\). A particle of mass \(m \mathrm {~kg}\) is released from rest at a point \(A\) on the sphere, such that angle \(T O A\) is \(\frac { \pi } { 6 }\) radians. The particle leaves the surface of the sphere at \(B\) (see diagram).

1. Show that \(\cos T O B = \frac { \sqrt { 3 } } { 3 }\).
2. Find the speed of the particle at \(B\).
3. Find the transverse acceleration of the particle at \(B\).

2

1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e0e5580a-e1f0-46f8-9304-2a96533af186-03_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Express \(v ^ { 2 }\) in terms of \(\theta\).
   2. Show that \(R = 5.88 \sin \theta - 3.43\).
   3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
   4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.

2

1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-3_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Express \(v ^ { 2 }\) in terms of \(\theta\).
   2. Show that \(R = 5.88 \sin \theta - 3.43\).
   3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
   4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.

2

1. A fixed solid sphere with a smooth surface has centre O and radius 0.8 m . A particle P is given a horizontal velocity of \(1.2 \mathrm {~ms} ^ { - 1 }\) at the highest point on the sphere, and it moves on the surface of the sphere in part of a vertical circle of radius 0.8 m .
   1. Find the radial and tangential components of the acceleration of P at the instant when OP makes an angle \(\frac { 1 } { 6 } \pi\) radians with the upward vertical. (You may assume that P is still in contact with the sphere.)
   2. Find the speed of P at the instant when it leaves the surface of the sphere.
2. Two fixed points R and S are 2.5 m apart with S vertically below R . A particle Q of mass 0.9 kg is connected to R and to S by two light inextensible strings; Q is moving in a horizontal circle at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with both strings taut. The radius of the circle is 2.4 m and the centre C of the circle is 0.7 m vertically below S, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3f674569-7e99-4ba8-84f1-a1eb438e30ed-2_547_720_1946_644} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Find the tension in the string RQ and the tension in the string \(S Q\).

1 A fixed solid sphere has centre O and radius 2.6 m . A particle P of mass 0.65 kg moves on the smooth surface of the sphere. The particle P is set in motion with horizontal velocity \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the highest point of the sphere, and moves in part of a vertical circle. When OP makes an angle \(\theta\) with the upward vertical, and P is still in contact with the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 52.92 - 50.96 \cos \theta\).
2. Find, in terms of \(\theta\), the normal reaction acting on P .
3. Find the speed of P at the instant when it leaves the surface of the sphere. The particle P is now attached to one end of a light inextensible string, and the other end of the string is fixed to a point A , vertically above O , such that AP is tangential to the sphere, as shown in Fig. 1. P moves with constant speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with radius 2.4 m on the surface of the sphere. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3ec81c4e-e0fa-43d9-9c79-ef9df746be8f-2_1100_634_1089_753} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
4. Find the tension in the string and the normal reaction acting on P .

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is at rest at the highest point \(A\) of a smooth sphere, centre \(O\), of radius 1.25 m which is fixed to a horizontal surface. When \(P\) is slightly disturbed it slides along the surface of the sphere. Whilst \(P\) is in contact with the sphere it has speed \(v \mathrm {~ms} ^ { - 1 }\) when \(\angle A O P = \theta\) as shown in Figure 1.

1. Show that \(v ^ { 2 } = 24.5 ( 1 - \cos \theta )\).
2. Find the value of \(\cos \theta\) when \(P\) leaves the surface of the sphere.

6 A smooth solid hemisphere of radius \(a\) is fixed with its plane face in contact with a horizontal surface.
The highest point on the hemisphere is H , and the centre of its base is O . A particle of mass \(m\) is held at a point S on the surface of the hemisphere such that angle HOS is \(30 ^ { \circ }\), as shown in Fig. 6. The particle is projected from S with speed \(0.8 \sqrt { a g }\) along the surface of the hemisphere towards H . \begin{figure}[h] \end{figure}

1. Show that the particle passes through H without leaving the surface of the hemisphere. After passing through H , the particle passes through a point Q on the surface of the hemisphere, where angle \(\mathrm { HOQ } = \theta ^ { \circ }\).
2. State, in terms of \(g\) and \(\theta\), the tangential component of the acceleration of the particle when it is at Q . The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L .
3. Find the value of \(\cos \theta\) correct to 3 significant figures.
4. Show that \(\mathrm { OL } = k a\), where \(k\) is to be found correct to 3 significant figures.

4 A fixed smooth sphere has centre O and radius \(a\). A particle P of mass \(m\) is placed at the highest point of the sphere and given an initial horizontal speed \(u\). For the first part of its motion, P remains in contact with the sphere and has speed \(v\) when OP makes an angle \(\theta\) with the upward vertical. This is shown in Fig. 4. \begin{figure}[h] \end{figure}

1. By considering the energy of P , show that \(v ^ { 2 } = u ^ { 2 } + 2 g a ( 1 - \cos \theta )\).
2. Show that the magnitude of the normal contact force between the sphere and particle P is $$m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a } .$$ The particle loses contact with the sphere when \(\cos \theta = \frac { 3 } { 4 }\).
3. Find an expression for \(u\) in terms of \(a\) and \(g\).