Questions — OCR M3 (138 questions)

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OCR M3 2015 June Q2
8 marks Standard +0.3
\includegraphics{figure_2} Two uniform rods \(AB\) and \(BC\), each of length \(2L\), are freely jointed at \(B\), and \(AB\) is freely jointed to a fixed point at \(A\). The rods are held in equilibrium in a vertical plane by a light horizontal string attached at \(C\). The rods \(AB\) and \(BC\) make angles \(\alpha\) and \(\beta\) to the horizontal respectively. The weight of rod \(BC\) is \(75\) N, and the tension in the string is \(50\) N (see diagram).
  1. Show that \(\tan \beta = \frac{1}{3}\). [3]
  2. Given that \(\tan \alpha = \frac{12}{5}\), find the weight of \(AB\). [5]
OCR M3 2015 June Q3
13 marks Challenging +1.2
\includegraphics{figure_3} A small object \(P\) is attached to one end of each of two vertical light elastic strings. One string is of natural length \(0.4\) m and has modulus of elasticity \(10\) N; the other string is of natural length \(0.5\) m and has modulus of elasticity \(12\) N. The upper ends of both strings are attached to a fixed horizontal beam and \(P\) hangs in equilibrium \(0.6\) m below the beam (see diagram).
  1. Show that the weight of \(P\) is \(7.4\) N and find the total elastic potential energy stored in the two strings when \(P\) is hanging in equilibrium. [6]
\(P\) is then held at a point \(0.7\) m below the beam with the strings vertical. \(P\) is released from rest.
  1. Show that, throughout the subsequent motion, \(P\) performs simple harmonic motion, and find the period. [7]
OCR M3 2015 June Q4
11 marks Standard +0.8
A particle of mass \(0.4\) kg, moving on a smooth horizontal surface, passes through a point \(O\) with velocity \(10\text{ ms}^{-1}\). At time \(t\) s after the particle passes through \(O\), the particle has a displacement \(x\) m from \(O\), has a velocity \(v\text{ ms}^{-1}\) away from \(O\), and is acted on by a force of magnitude \(\frac{1}{5}v\) N acting towards \(O\). Find
  1. the time taken for the velocity of the particle to reduce from \(10\text{ ms}^{-1}\) to \(5\text{ ms}^{-1}\), [5]
  2. the average velocity of the particle over this time. [6]
OCR M3 2015 June Q5
11 marks Challenging +1.2
\includegraphics{figure_5} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2m\) kg and \(m\) kg respectively. The spheres are moving on a horizontal surface when they collide. Before the collision, \(A\) is moving with speed \(a\text{ ms}^{-1}\) in a direction making an angle \(\alpha\) with the line of centres and \(B\) is moving towards \(A\) with speed \(b\text{ ms}^{-1}\) in a direction making an angle \(\beta\) with the line of centres (see diagram). After the collision, \(A\) moves with velocity \(2\text{ ms}^{-1}\) in a direction perpendicular to the line of centres and \(B\) moves with velocity \(2\text{ ms}^{-1}\) in a direction making an angle of \(45°\) with the line of centres. The coefficient of restitution between \(A\) and \(B\) is \(\frac{2}{3}\).
  1. Show that \(a\cos \alpha = \frac{5}{3}\sqrt{2}\) and find \(b\cos \beta\). [7]
  2. Find the values of \(a\) and \(\alpha\). [4]
OCR M3 2015 June Q6
11 marks Standard +0.3
A particle \(P\) starts from rest from a point \(A\) and moves in a straight line with simple harmonic motion about a point \(O\). At time \(t\) seconds after the motion starts the displacement of \(P\) from \(O\) is \(x\) m towards \(A\). The particle \(P\) is next at rest when \(t = 0.25\pi\) having travelled a distance of \(1.2\) m.
  1. Find the maximum velocity of \(P\). [3]
  2. Find the value of \(x\) and the velocity of \(P\) when \(t = 0.7\). [4]
  3. Find the other values of \(t\), for \(0 < t < 1\), at which \(P\)'s speed is the same as when \(t = 0.7\). Find also the corresponding values of \(x\). [4]
OCR M3 2015 June Q7
12 marks Standard +0.3
\includegraphics{figure_7} One end of a light inextensible string of length \(0.5\) 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 projected horizontally from the point \(0.5\) m below \(O\) with speed \(u\text{ ms}^{-1}\). When the string makes an angle of \(\theta\) with the downward vertical the particle has speed \(v\text{ ms}^{-1}\) (see diagram).
  1. Show that, while the string is taut, the tension, \(T\) N, in the string is given by $$T = 5.88\cos \theta + 0.4u^2 - 3.92.$$ [5]
  2. Find the least value of \(u\) for which the particle will move in a complete circle. [3]
  3. If in fact \(u = 3.5\text{ ms}^{-1}\), find the speed of the particle at the point where the string first becomes slack. [4]
END OF QUESTION PAPER
OCR M3 2016 June Q1
6 marks Standard +0.3
\includegraphics{figure_1} A particle \(P\) of mass \(0.3\) kg is moving with speed \(0.4\) m s\(^{-1}\) in a straight line on a smooth horizontal surface when it is struck by a horizontal impulse. After the impulse acts \(P\) has speed \(0.6\) m s\(^{-1}\) and is moving in a direction making an angle \(30°\) with its original direction of motion (see diagram).
  1. Find the magnitude of the impulse and the angle its line of action makes with the original direction of motion of \(P\). [4]
Subsequently a second impulse acts on \(P\). After this second impulse acts, \(P\) again moves from left to right with speed \(0.4\) m s\(^{-1}\) in a direction parallel to its original direction of motion.
  1. State the magnitude of the second impulse, and show the direction of the second impulse on a diagram. [2]
OCR M3 2016 June Q2
8 marks Standard +0.3
A particle \(Q\) of mass \(0.2\) kg is projected horizontally with velocity \(4\) m s\(^{-1}\) from a fixed point \(A\) on a smooth horizontal surface. At time \(t\) s after projection \(Q\) is \(x\) m from \(A\) and is moving away from \(A\) with velocity \(v\) m s\(^{-1}\). There is a force of \(3\cos 2t\) N acting on \(Q\) in the positive \(x\)-direction.
  1. Find an expression for the velocity of \(Q\) at time \(t\). State the maximum and minimum values of the velocity of \(Q\) as \(t\) varies. [4]
  2. Find the average velocity of \(Q\) between times \(t = \pi\) and \(t = \frac{3}{2}\pi\). [4]
OCR M3 2016 June Q3
10 marks Challenging +1.2
\includegraphics{figure_3} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2m\) kg and \(3m\) kg respectively. The spheres are approaching each other on a horizontal surface when they collide. Before the collision \(A\) is moving with speed \(5\) m s\(^{-1}\) in a direction making an angle \(\alpha\) with the line of centres, where \(\cos \alpha = \frac{4}{5}\), and \(B\) is moving with speed \(3\frac{1}{4}\) m s\(^{-1}\) in a direction making an angle \(\beta\) with the line of centres, where \(\cos \beta = \frac{5}{13}\). A straight vertical wall is situated to the right of \(B\), perpendicular to the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is \(\frac{2}{5}\).
  1. 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. [7]
\(B\) subsequently hits the wall.
  1. 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. [3]
OCR M3 2016 June Q4
7 marks Challenging +1.8
\includegraphics{figure_4} \(A\) 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 \(ABC\) forms a straight line and \(AB = 0.4\) m (see diagram). Find the greatest kinetic energy of particle \(B\) in the subsequent motion. [7]
OCR M3 2016 June Q5
11 marks Challenging +1.2
\includegraphics{figure_5} 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 \(2\sqrt{ag}\). When the string makes an angle \(\theta\) with the upward vertical \(P\) has speed \(v\) (see diagram). The tension in the string is \(T\).
  1. 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. [6]
\(P\) is now projected horizontally from the same initial position with speed \(U\).
  1. Find the set of values of \(U\) for which the string does not remain taut in the subsequent motion. [5]
OCR M3 2016 June Q6
13 marks Standard +0.8
\includegraphics{figure_6} Two uniform rods \(AB\) and \(AC\) are freely jointed at \(A\). Rod \(AB\) is of length \(2l\) and weight \(W\); rod \(AC\) is of length \(4l\) and weight \(2W\). The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that \(AB\) makes an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac{3}{5}\), and \(AC\) makes an angle of \(\varphi\) with the horizontal, where \(\sin \varphi = \frac{1}{5}\) (see diagram). The force of the step acting on \(AB\) at \(B\) has vertical component \(R\) and horizontal component \(F\).
  1. By taking moments about \(A\) for the rod \(AB\), find an equation relating \(W\), \(R\) and \(F\). [3]
  2. Show that \(R = \frac{75}{68}W\), and find the vertical component of the force acting on \(AC\) at \(C\). [6]
  3. 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. [4]
OCR M3 2016 June Q7
17 marks Challenging +1.2
A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of modulus of elasticity \(24mg\) N and natural length \(0.6\) m. The other end of the string is attached to a fixed point \(O\); the particle \(P\) rests in equilibrium at a point \(A\) with the string vertical.
  1. Show that the distance \(OA\) is \(0.625\) m. [2]
Another particle \(Q\), of mass \(3m\) kg, is released from rest from a point \(0.4\) m above \(P\) and falls onto \(P\). The two particles coalesce.
  1. Show that the combined particle initially moves with speed \(2.1\) m s\(^{-1}\). [3]
  2. Show that the combined particle initially performs simple harmonic motion, and find the centre of this motion and its amplitude. [5]
  3. Find the time that elapses between \(Q\) being released from rest and the combined particle first reaching the highest point of its subsequent motion. [7]