CAIE M1 (Mechanics 1) 2017 November

1 A particle of mass 0.2 kg is resting in equilibrium on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal.
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1. Show that the friction force acting on the particle is 0.684 N , correct to 3 significant figures. [1]
   The coefficient of friction between the particle and the plane is 0.6 . A force of magnitude 0.9 N is applied to the particle down a line of greatest slope of the plane. The particle accelerates down the plane.
2. Find this acceleration.
   \includegraphics[max width=\textwidth, alt={}, center]{d3bb18c3-8c6f-41d4-808e-479f0c92250f-03_250_657_258_744} A block of mass 15 kg hangs in equilibrium below a horizontal ceiling attached to two strings as shown in the diagram. One of the strings is inclined at \(45 ^ { \circ }\) to the horizontal and the tension in this string is 120 N . The other string is inclined at \(\theta ^ { \circ }\) to the horizontal and the tension in this string is \(T \mathrm {~N}\). Find the values of \(T\) and \(\theta\).

3 A car travels along a straight road with constant acceleration. It passes through points \(A , B\) and \(C\). The car passes point \(A\) with velocity \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The two sections \(A B\) and \(B C\) are of equal length. The times taken to travel along \(A B\) and \(B C\) are 5 s and 3 s respectively.

1. Write down an expression for the distance \(A B\) in terms of the acceleration of the car. Write down a similar expression for the distance \(A C\). Hence show that the acceleration of the car is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
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   \includegraphics[max width=\textwidth, alt={}, center]{d3bb18c3-8c6f-41d4-808e-479f0c92250f-04_67_1575_580_324}
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2. Find the speed of the car as it passes point \(C\).

4 A particle \(P\) is projected vertically upwards from horizontal ground with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the time taken for \(P\) to return to the ground.
   The time in seconds after \(P\) is projected is denoted by \(t\). When \(t = 1\), a second particle \(Q\) is projected vertically upwards with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point which is 5 m above the ground. Particles \(P\) and \(Q\) move in different vertical lines.
2. Find the set of values of \(t\) for which the two particles are moving in the same direction.

5 A cyclist is riding up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.04\). The total mass of the bicycle and rider is 80 kg . The cyclist is riding at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a force resisting the motion. The work done by the cyclist against this resistance force over a distance of 25 m is 600 J .

1. Find the power output of the cyclist.
   The cyclist reaches the top of the hill, where the road becomes horizontal, with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist continues to work at the same rate on the horizontal part of the road.
2. Find the speed of the cyclist 10 seconds after reaching the top of the hill, given that the work done by the cyclist during this period against the resistance force is 1200 J .
   \includegraphics[max width=\textwidth, alt={}, center]{d3bb18c3-8c6f-41d4-808e-479f0c92250f-08_346_616_260_762} Two particles \(P\) and \(Q\), each of mass \(m \mathrm {~kg}\), are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 7 } { 24 }\). Particle \(P\) rests on the plane and particle \(Q\) hangs vertically, as shown in the diagram. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. The system is in limiting equilibrium.

1. Show that the coefficient of friction between \(P\) and the plane is \(\frac { 4 } { 3 }\).
   A force of magnitude 10 N is applied to \(P\), acting up a line of greatest slope of the plane, and \(P\) accelerates at \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(m\).

7 A particle starts from rest and moves in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after the start is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = - 0.01 t ^ { 3 } + 0.22 t ^ { 2 } - 0.4 t$$

1. Find the two positive values of \(t\) for which the particle is instantaneously at rest.
2. Find the time at which the acceleration of the particle is greatest.
3. Find the distance travelled by the particle while its velocity is positive.