OCR M1 (Mechanics 1) 2012 June

1
\includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-2_305_295_264_868} Two perpendicular forces of magnitudes \(F \mathrm {~N}\) and 8 N act at a point \(O\) (see diagram). Their resultant has magnitude 17 N .

1. Calculate \(F\) and find the angle which the resultant makes with the 8 N force. A third force of magnitude \(E \mathrm {~N}\), acting in the same plane as the two original forces, is now applied at the point \(O\). The three forces of magnitudes \(E N , F N\) and \(8 N\) are in equilibrium.
2. State the value of \(E\) and the angle between the directions of the \(E \mathrm {~N}\) and 8 N forces.

2 A particle is projected vertically upwards with speed \(7 \mathrm {~ms} ^ { - 1 }\) from a point on the ground.

1. Find the speed of the particle and its distance above the ground 0.4 s after projection.
2. Find the total distance travelled by the particle in the first 0.9 s after projection.

3
\(\mathrm { v } \left( \mathrm { ms } ^ { - 1 } \right)\)
\includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-2_449_1121_1500_440}
not to scale The diagram shows the \(( t , v )\) graphs for two athletes, \(A\) and \(B\), who run in the same direction in the same straight line while they exchange the baton in a relay race. \(A\) runs with constant velocity \(10 \mathrm {~ms} ^ { - 1 }\) until he decelerates at \(5 \mathrm {~ms} ^ { - 2 }\) and subsequently comes to rest. \(B\) has constant acceleration from rest until reaching his constant speed of \(10 \mathrm {~ms} ^ { - 1 }\). The baton is exchanged 2 s after \(B\) starts running, when both athletes have speed \(8 \mathrm {~ms} ^ { - 1 }\) and \(B\) is 1 m ahead of \(A\).

1. Find the value of \(t\) at which \(A\) starts to decelerate.
2. Calculate the distance between \(A\) and \(B\) at the instant when \(B\) starts to run.

4 A block \(B\) of weight 28 N is pulled at constant speed across a rough horizontal surface by a force of magnitude 14 N inclined at \(30 ^ { \circ }\) above the horizontal.

1. Show that the coefficient of friction between the block and the surface is 0.577 , correct to 3 significant figures. The 14 N force is suddenly removed, and the block decelerates, coming to rest after travelling a further 3.2 m .
2. Calculate the speed of the block at the instant the 14 N force was removed.

5
\includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-3_291_182_799_945} Particles \(P\) and \(Q\), of masses 0.4 kg and \(m \mathrm {~kg}\) respectively, are joined by a light inextensible string which passes over a smooth pulley. The particles are released from rest at the same height above a horizontal surface; the string is taut and the portions of the string not in contact with the pulley are vertical (see diagram). \(Q\) begins to descend with acceleration \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and reaches the surface 0.3 s after being released. Subsequently, \(Q\) remains at rest and \(P\) never reaches the pulley.

1. Calculate the tension in the string while \(Q\) is in motion.
2. Calculate the momentum lost by \(Q\) when it reaches the surface.
3. Calculate the greatest height of \(P\) above the surface. \section*{[Questions 6 and 7 are printed overleaf.]}

6
\includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_328_698_255_657} A particle \(P\) lies on a slope inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) is attached to one end of a taut light inextensible string which passes through a small smooth ring \(Q\) of mass \(m \mathrm {~kg}\). The portion \(P Q\) of the string is horizontal and the other portion of the string is inclined at \(40 ^ { \circ }\) to the vertical. A horizontal force of magnitude \(H \mathrm {~N}\), acting away from \(P\), is applied to \(Q\) (see diagram). The tension in the string is 6.4 N , and the string is in the vertical plane containing the line of greatest slope on which \(P\) lies. Both \(P\) and \(Q\) are in equilibrium.

1. Calculate \(m\).
2. Calculate \(H\).
3. Given that the weight of \(P\) is 32 N , and that \(P\) is in limiting equilibrium, show that the coefficient of friction between \(P\) and the slope is 0.879 , correct to 3 significant figures.
   \(Q\) and the string are now removed.
4. Determine whether \(P\) remains in equilibrium.

7
\includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_122_255_1503_561} The diagram shows two particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, which move on a horizontal surface in the same direction along a straight line. A stationary particle \(R\) of mass 1.5 kg also lies on this line. \(P\) and \(Q\) collide and coalesce to form a combined particle \(C\). Immediately before this collision \(P\) has velocity \(4 \mathrm {~ms} ^ { - 1 }\) and \(Q\) has velocity \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the velocity of \(C\) immediately after this collision. At time \(t \mathrm {~s}\) after this collision the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(C\) is given by \(v = V _ { 0 } - 3 t ^ { 2 }\) for \(0 < t \leqslant 0.3\). \(C\) strikes \(R\) when \(t = 0.3\).
2. (a) State the value of \(V _ { 0 }\).
   (b) Calculate the distance \(C\) moves before it strikes \(R\).
   (c) Find the acceleration of \(C\) immediately before it strikes \(R\). Immediately after \(C\) strikes \(R\), the particles have equal speeds but move in opposite directions.
3. Find the speed of \(C\) immediately after it strikes \(R\).