Questions — OCR M3 (130 questions)

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OCR M3 2010 January Q6
  1. By considering the total energy of the system, obtain an expression for \(v ^ { 2 }\) in terms of \(\theta\).
  2. Show that the magnitude of the force exerted on \(P\) by the cylinder is \(( 7.12 \sin \theta - 4.64 \theta ) \mathrm { N }\).
  3. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(1.53 < \alpha < 1.54\).
OCR M3 2007 June Q6
  1. Show that, when \(P\) is in equilibrium, \(O P = 7.25 \mathrm {~m}\).
  2. Verify that \(P\) and \(Q\) together just reach the safety net.
  3. At the lowest point of their motion \(P\) releases \(Q\). Prove that \(P\) subsequently just reaches \(O\).
  4. State two additional modelling assumptions made when answering this question.
OCR M3 2010 June Q3
  1. Given that the coefficient of restitution between the sphere and the wall is \(\frac { 1 } { 2 }\), state the values of \(u\) and \(v\). Shortly after hitting the wall the sphere \(A\) comes into contact with another uniform smooth sphere \(B\), which has the same mass and radius as \(A\). The sphere \(B\) is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-3_524_371_1503_888} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. Find, for each sphere, its speed and its direction of motion immediately after the contact.
    \(4 ~ O\) is a fixed point on a horizontal plane. A particle \(P\) of mass 0.25 kg is released from rest at \(O\) and moves in a straight line on the plane. At time \(t \mathrm {~s}\) after release the only horizontal force acting on \(P\) has magnitude $$\frac { 1 } { 2400 } \left( 144 - t ^ { 2 } \right) \mathrm { N } \quad \text { for } 0 \leqslant t \leqslant 12$$ and $$\frac { 1 } { 2400 } \left( t ^ { 2 } - 144 \right) \mathrm { N } \text { for } t \geqslant 12 .$$ The force acts in the direction of \(P\) 's motion. \(P\) 's velocity at time \(t \mathrm {~s}\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find an expression for \(v\) in terms of \(t\), valid for \(t \geqslant 12\), and hence show that \(v\) is three times greater when \(t = 24\) than it is when \(t = 12\).
  4. Sketch the \(( t , v )\) graph for \(0 \leqslant t \leqslant 24\).
OCR M3 2016 June Q3
  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.
    \(B\) subsequently hits the wall.
  2. 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. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{4} \includegraphics[alt={},max width=\textwidth]{c0f31235-80aa-4838-844f-b706de55e7cd-3_193_1451_705_306}
    \end{figure} \(A\) 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 \(A B C\) forms a straight line and \(A B = 0.4 \mathrm {~m}\) (see diagram). Find the greatest kinetic energy of particle \(B\) in the subsequent motion.
    \includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-3_586_533_1409_758} 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 \(2 \sqrt { a g }\). When the string makes an angle \(\theta\) with the upward vertical \(P\) has speed \(v\) (see diagram). The tension in the string is \(T\).
  3. 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.
    \(P\) is now projected horizontally from the same initial position with speed \(U\).
  4. Find the set of values of \(U\) for which the string does not remain taut in the subsequent motion.
    \includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-4_566_1013_255_525} Two uniform rods \(A B\) and \(A C\) are freely jointed at \(A\). Rod \(A B\) is of length \(2 l\) and weight \(W\); \(\operatorname { rod } A C\) is of length \(4 l\) and weight \(2 W\). The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that \(A B\) makes an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac { 4 } { 5 }\), and \(A C\) makes an angle of \(\varphi\) with the horizontal, where \(\sin \varphi = \frac { 3 } { 5 }\) (see diagram). The force of the step acting on \(A B\) at \(B\) has vertical component \(R\) and horizontal component \(F\).
  5. By taking moments about \(A\) for the rod \(A B\), find an equation relating \(W , R\) and \(F\).
  6. Show that \(R = \frac { 73 } { 50 } W\), and find the vertical component of the force acting on \(A C\) at \(C\).
  7. 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.
OCR M3 2006 January Q10
10 JANUARY 2006 Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
List of Formulae (MF1) TIME
1 hour 30 minutes
  • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
  • Answer all the questions.
  • Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
  • The acceleration due to gravity is denoted by \(\mathrm { g } \mathrm { m } \mathrm { s } ^ { - 2 }\). Unless otherwise instructed, when a numerical value is needed, use \(g = 9.8\).
  • You are permitted to use a graphical calculator in this paper.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total number of marks for this paper is 72.
  • Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
  • You are reminded of the need for clear presentation in your answers.