5
\includegraphics[max width=\textwidth, alt={}, center]{4799bae3-8d1e-4ca7-8621-6fa195469175-4_225_1239_367_324}
The masses of two spheres \(A\) and \(B\) are \(3 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving towards each other with constant speeds \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively along the same straight line towards each other on a smooth horizontal surface (see diagram). The two spheres collide and the coefficient of restitution between the spheres is \(e\). After colliding, \(A\) and \(B\) both move in the same direction with speeds \(v \mathrm {~ms} ^ { - 1 }\) and \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively.
- Find an expression for \(v\) in terms of \(e\) and \(u\).
- Write down unsimplified expressions in terms of \(e\) and \(u\) for
(a) the total kinetic energy of the spheres before the collision,
(b) the total kinetic energy of the spheres after the collision. - Given that the total kinetic energy of the spheres after the collision is \(\lambda\) times the total kinetic energy before the collision, show that
$$\lambda = \frac { 27 e ^ { 2 } + 25 } { 52 }$$
- Comment on the cases when
(a) \(\lambda = 1\),
(b) \(\lambda = \frac { 25 } { 52 }\).
\includegraphics[max width=\textwidth, alt={}, center]{4799bae3-8d1e-4ca7-8621-6fa195469175-5_808_990_319_539}
The fixed points \(A\), \(B\) and \(C\) are in a vertical line with \(A\) above \(B\) and \(B\) above \(C\). A particle \(P\) of mass 2.5 kg is joined to \(A\), to \(B\) and to a particle \(Q\) of mass 2 kg , by three light rods where the length of rod \(A P\) is 1.5 m and the length of rod \(P Q\) is 0.75 m . Particle \(P\) moves in a horizontal circle with centre \(B\). Particle \(Q\) moves in a horizontal circle with centre \(C\) at the same constant angular speed \(\omega\) as \(P\), in such a way that \(A , B , P\) and \(Q\) are coplanar. The rod \(A P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical, rod \(P Q\) makes an angle of \(30 ^ { \circ }\) with the downward vertical and rod \(B P\) is horizontal (see diagram). - Find the tension in the \(\operatorname { rod } P Q\).
- Find \(\omega\).
- Find the speed of \(P\).
- Find the tension in the rod \(A P\).
- Hence find the magnitude of the force in rod \(B P\).
Decide whether this rod is under tension or compression.
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