Find the speed of \(A\) after the collision. Find also the component of the velocity of \(B\) along the line of centres after the collision.
\(B\) subsequently hits the wall.
Explain why \(A\) and \(B\) will have a second collision if the coefficient of restitution between \(B\) and the wall is sufficiently large. Find the set of values of the coefficient of restitution for which this second collision will occur.
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\(A\) and \(C\) are two fixed points, 1.5 m apart, on a smooth horizontal plane. A light elastic string of natural length 0.4 m and modulus of elasticity 20 N has one end fixed to point \(A\) and the other end fixed to a particle \(B\). Another light elastic string of natural length 0.6 m and modulus of elasticity 15 N has one end fixed to point \(C\) and the other end fixed to the particle \(B\). The particle is released from rest when \(A B C\) forms a straight line and \(A B = 0.4 \mathrm {~m}\) (see diagram).
Find the greatest kinetic energy of particle \(B\) in the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-3_586_533_1409_758}
One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs at rest. \(P\) is then projected horizontally from this position with speed \(2 \sqrt { a g }\). When the string makes an angle \(\theta\) with the upward vertical \(P\) has speed \(v\) (see diagram). The tension in the string is \(T\).
Find an expression for \(T\) in terms of \(m , g\) and \(\theta\), and hence find the height of \(P\) above its initial level when the string becomes slack.
\(P\) is now projected horizontally from the same initial position with speed \(U\).
Find the set of values of \(U\) for which the string does not remain taut in the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-4_566_1013_255_525}
Two uniform rods \(A B\) and \(A C\) are freely jointed at \(A\). Rod \(A B\) is of length \(2 l\) and weight \(W\); \(\operatorname { rod } A C\) is of length \(4 l\) and weight \(2 W\). The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that \(A B\) makes an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac { 4 } { 5 }\), and \(A C\) makes an angle of \(\varphi\) with the horizontal, where \(\sin \varphi = \frac { 3 } { 5 }\) (see diagram). The force of the step acting on \(A B\) at \(B\) has vertical component \(R\) and horizontal component \(F\).
By taking moments about \(A\) for the rod \(A B\), find an equation relating \(W , R\) and \(F\).
Show that \(R = \frac { 73 } { 50 } W\), and find the vertical component of the force acting on \(A C\) at \(C\).
The coefficient of friction at \(B\) is equal to that at \(C\). Given that one of the rods is on the point of slipping, explain which rod this must be, and find the coefficient of friction.