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\includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_339_511_349_817}
A particle of mass 0.3 kg is attached to one end \(A\) of a light inextensible string of length 1.5 m . The other end \(B\) of the string is attached to a ceiling, so that the particle may swing in a vertical plane. The particle is released from rest when the string is taut and makes an angle of \(75 ^ { \circ }\) with the vertical (see diagram). Air resistance may be regarded as being negligible.
- Show that, at an instant when the string makes an angle of \(40 ^ { \circ }\) with the vertical, the speed of the particle is \(3.90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
- By considering Newton's second law, along and perpendicular to the string, find the radial and transverse components of acceleration, at this same instant, and hence the magnitude of the acceleration of the particle.
\includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_419_604_1370_772}
A smooth sphere of mass 0.3 kg is moving in a straight line on a horizontal surface. It collides with a vertical wall when the velocity of the sphere is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the wall (see diagram). The coefficient of restitution between the sphere and the wall is 0.4 .
- (a) Find the component of the velocity of the sphere perpendicular to the wall immediately after the collision.
(b) Find the magnitude of the impulse exerted by the wall on the sphere. - Determine the magnitude and direction of the velocity of the sphere immediately after the collision, giving the direction as an acute angle to the wall.