7.
\includegraphics[max width=\textwidth, alt={}]{4718e448-71e5-452a-a905-608331f743a5-5_520_410_429_228}
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).
- Find the speed of \(B\) at the instant when \(O B\) makes an angle of 0.8 radians with the downward vertical through \(O\).
- Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\).
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As shown in the diagram, \(A B\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length 1 m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m _ { 1 } \mathrm {~kg}\). One end of another light inextensible string of length 1 m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m _ { 2 } \mathrm {~kg}\), which is free to move on \(A B\).
Initially, \(P\) moves in a horizontal circle of radius 0.6 m with constant angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). The magnitude of the tension in string \(A P\) is denoted by \(T _ { 1 } \mathrm {~N}\) while that in string \(P R\) is denoted by \(T _ { 2 } \mathrm {~N}\). - By considering forces on \(R\), express \(T _ { 2 }\) in terms of \(m _ { 2 }\).
- Show that
- \(T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)\),
- \(\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }\).
- Deduce that, in the case where \(m _ { 1 }\) is much bigger than \(m _ { 2 } , \omega \approx 3.5\).
In a different case, where \(m _ { 1 } = 2.5\) and \(m _ { 2 } = 2.8 , P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
- Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm { rads } ^ { - 1 }\) to zero.
Three particles, \(P , Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(k m\) respectively, where \(k\) is a constant.
Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly.
The coefficient of restitution between each pair of particles is \(e\). - Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision.
Given that the mass of \(P\) is \(k m\) and that there is a second collision,
- write down, in terms of \(u , k\) and \(e\), the speed of \(Q\) after this second collision.