Questions — OCR Further Mechanics AS (53 questions)

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OCR Further Mechanics AS Specimen Q4
10 marks Standard +0.8
A car of mass 1250 kg experiences a resistance to its motion of magnitude \(kv^2\) N, where \(k\) is a constant and \(v \, \text{m s}^{-1}\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P\) W. At a point \(A\) on the road the car's speed is \(15 \, \text{m s}^{-1}\) and it has an acceleration of magnitude \(0.54 \, \text{m s}^{-2}\). At a point \(B\) on the road the car's speed is \(20 \, \text{m s}^{-1}\) and it has an acceleration of magnitude \(0.3 \, \text{m s}^{-2}\).
  1. Find the values of \(k\) and \(P\). [7]
The power is increased to 15 kW.
  1. Calculate the maximum steady speed of the car on a straight horizontal road. [3]
OCR Further Mechanics AS Specimen Q5
15 marks Standard +0.8
\includegraphics{figure_5} The masses of two spheres \(A\) and \(B\) are \(3m\) kg and \(m\) kg respectively. The spheres are moving towards each other with constant speeds \(2u \, \text{m s}^{-1}\) and \(u \, \text{m s}^{-1}\) 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 \(v \, \text{m s}^{-1}\) and \(w \, \text{m s}^{-1}\), respectively.
  1. Find an expression for \(v\) in terms of \(e\) and \(u\). [6]
  2. Write down unsimplified expressions in terms of \(e\) and \(u\) for
    1. the total kinetic energy of the spheres before the collision, [1]
    2. the total kinetic energy of the spheres after the collision. [2]
  3. Given that the total kinetic energy of the spheres after the collision is \(\lambda\) times the total kinetic energy before the collision, show that $$\lambda = \frac{27e^2 + 25}{52}.$$ [3]
  4. Comment on the cases when
    1. \(\lambda = 1\),
    2. \(\lambda = \frac{25}{52}\). [3]
OCR Further Mechanics AS Specimen Q6
13 marks Challenging +1.2
\includegraphics{figure_6} 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 \(AP\) is 1.5 m and the length of rod \(PQ\) 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 \(AP\) makes an angle of \(60°\) with the downward vertical, rod \(PQ\) makes an angle of \(30°\) with the downward vertical and rod \(BP\) is horizontal (see diagram).
  1. Find the tension in the rod \(PQ\). [2]
  2. Find \(\omega\). [3]
  3. Find the speed of \(P\). [1]
  4. Find the tension in the rod \(AP\). [3]
  5. Hence find the magnitude of the force in rod \(BP\). Decide whether this rod is under tension or compression. [4]