6.03h Variable force impulse: using integration

4 \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_512_983_267_580} A uniform sphere rests on a horizontal plane. The sphere has centre \(O\), radius 0.6 m and weight 36 N . A uniform rod \(A B\), of weight 14 N and length 1 m , rests with \(A\) in contact with the plane and \(B\) in contact with the sphere at the end of a horizontal diameter. The point of contact of the sphere with the plane is \(C\), and \(A , B , C\) and \(O\) lie in the same vertical plane (see diagram). The contacts at \(A , B\) and \(C\) are rough and the system is in equilibrium. By taking moments about \(C\) for the system, show that the magnitude of the normal contact force at \(A\) is 10 N . Show that the magnitudes of the frictional forces at \(A , B\) and \(C\) are equal. The coefficients of friction at \(A , B\) and \(C\) are all equal to \(\mu\). Find the smallest possible value of \(\mu\).

7 A disc, of mass 0.15 kg , slides across a smooth horizontal table and collides with a vertical wall which is perpendicular to the path of the disc. The disc is in contact with the wall for 0.02 seconds and then rebounds.
A possible model for the force, \(F\) newtons, exerted on the disc by the wall, whilst in contact, is given by $$F = k t ^ { 2 } ( t - b ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.020$$ where \(k\) and \(b\) are constants.
The force is initially zero and becomes zero again as the disc loses contact with the wall. 7

1. State the value of \(b\).
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2. Find the magnitude of the impulse on the disc, giving your answer in terms of \(k\).
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3. The disc is travelling at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the wall.
   The disc rebounds with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find \(k\).
   [0pt] [3 marks]

A ball of mass 0.15 kg is hit directly by a vertical cricket bat. Immediately before the impact, the ball is travelling horizontally with speed 28 m s\(^{-1}\) Immediately after the impact, the ball is travelling horizontally with speed 14 m s\(^{-1}\) in the opposite direction.

1. Find the magnitude of the impulse exerted by the bat on the ball. [2 marks]
2. In a simple model the force, \(F\) newtons, exerted by the bat on the ball, \(t\) seconds after the initial impact, is given by \(F = 10kt (0.05 - t)\) where \(k\) is a constant. Given the ball is in contact with the bat for 0.05 seconds, find the value of \(k\) [3 marks]

A single force, \(F\) newtons, acts on a particle moving on a straight, smooth, horizontal line. The force \(F\) acts in the direction of motion of the particle. At time \(t\) seconds, \(F = 6e^t + 2e^{2t}\) where \(0 \leq t \leq \ln 8\)

1. Find the impulse of \(F\) over the interval \(0 \leq t \leq \ln 8\) [2 marks]
2. The particle has a mass of 2 kg and at time \(t = 0\) has velocity \(5 \text{ m s}^{-1}\) Find the velocity of the particle when \(t = \ln 8\) [3 marks]