4 One end of a light inextensible string of length 2 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is held at rest with the string taut so that \(O P\) makes an angle of 0.15 radians with the downward vertical. \(P\) is released and \(t\) seconds afterwards \(O P\) makes an angle of \(\theta\) radians with the downward vertical.
- Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4.9 \sin \theta\) and give a reason why the motion is approximately simple harmonic.
Using the simple harmonic approximation,
- obtain an expression for \(\theta\) in terms of \(t\) and hence find the values of \(t\) at the first and second occasions when \(\theta = - 0.1\),
- find the angular speed of \(O P\) and the linear speed of \(P\) when \(t = 0.5\).
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Two uniform smooth identical spheres \(A\) and \(B\) are moving towards each other on a horizontal surface when they collide. Immediately before the collision \(A\) and \(B\) are moving with speeds \(u _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, at acute angles \(\alpha\) and \(\beta\), respectively, to the line of centres. Immediately after the collision \(A\) and \(B\) are moving with speeds \(v _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, at right angles and at acute angle \(\gamma\), respectively, to the line of centres (see diagram). - Given that \(\sin \beta = 0.96\) and \(\frac { v _ { B } } { u _ { B } } = 1.2\), find the value of \(\sin \gamma\).
- Given also that, before the collision, the component of \(A\) 's velocity parallel to the line of centres is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the values of \(u _ { B }\) and \(v _ { B }\).
- Find the coefficient of restitution between the spheres.
- Given that the kinetic energy of \(A\) immediately before the collision is \(6.5 m \mathrm {~J}\), where \(m \mathrm {~kg}\) is the mass of \(A\), find the value of \(v _ { A }\).