Edexcel M1 (Mechanics 1)

1. Briefly define the following terms used in modelling in Mechanics:
   1. lamina,
   2. uniform rod,
   3. smooth surface,
   4. particle.
      (4 marks)
   5. Two forces \(\mathbf { F }\) and \(\mathbf { G }\) are given by \(\mathbf { F } = ( 6 \mathbf { i } - 5 \mathbf { j } ) \mathbf { N } , \mathbf { G } = ( 3 \mathbf { i } + 17 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the \(x\) and \(y\) directions respectively and the unit of length on each axis is 1 cm .
      (a) Find the magnitude of \(\mathbf { R }\), the resultant of \(\mathbf { F }\) and \(\mathbf { G }\).
      (b) Find the angle between the direction of \(\mathbf { R }\) and the positive \(x\)-axis.
      \(\mathbf { R }\) acts through the point \(P ( - 4,3 )\). \(O\) is the origin \(( 0,0 )\).
      (c) Use the fact that \(O P\) is perpendicular to the line of action of \(\mathbf { R }\) to calculate the moment of \(\mathbf { R }\) about an axis through the origin and perpendicular to the \(x - y\) plane.
   6. A string is attached to a packing case of mass 12 kg , which is at rest on a rough horizontal plane. When a force of magnitude 50 N is applied at the other end of the string, and
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      the string makes an angle of \(35 ^ { \circ }\) with the vertical as shown, the case is on the point of moving.
      (a) Find the coefficient of friction between the case and the plane.
   The force is now increased, with the string at the same angle, and the case starts to move along the plane with constant acceleration, reaching a speed of \(2 \mathrm {~ms} ^ { - 1 }\) after 4 seconds.
   (b) Find the magnitude of the new force.
   (c) State any modelling assumptions you have made about the case and the string.

4. A uniform yoke \(A B\), of mass 4 kg and length \(4 a \mathrm {~m}\), rests on the shoulders \(S\) and \(T\) of two oxen.
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\(A S = T B = a \mathrm {~m}\). A bucket of mass \(x \mathrm {~kg}\) is suspended from \(A\).

1. Show that the vertical force on the yoke at \(T\) has magnitude \(\left( 2 - \frac { 1 } { 2 } x \right) g \mathrm {~N}\) and find, in terms of \(x\) and \(g\), the vertical force on the yoke at \(S\).
2. If the ratio of these vertical forces is \(5 : 1\), find the value of \(x\).
3. Find the maximum value of \(x\) for which the yoke will remain horizontal.

5. Two small smooth spheres \(A\) and \(B\), of equal radius but masses \(m \mathrm {~kg}\) and km kg respectively, where \(k > 1\), move towards each other along a straight line and collide directly. Immediately before the collision, \(A\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision, the impulse exerted by \(A\) on \(B\) has magnitude 7 km Ns.
Q. 5 continued on next page ... \section*{MECHANICS 1 (A) TEST PAPER 7 Page 2}

1. continued ...
   1. Find the speed of \(B\) after the impact.
   2. Show that the speed of \(A\) immediately after the collision is \(( 7 k - 5 ) \mathrm { ms } ^ { - 1 }\) and deduce that the direction of \(A\) 's motion is reversed.
      \(B\) is now given a further impulse of magnitude \(m u \mathrm { Ns }\), as a result of which a second collision between it and \(A\) occurs.
   3. Show that \(u > k ( 7 k - 1 )\).
   \includegraphics[max width=\textwidth, alt={}]{6fbe12d6-9a46-4602-a7b9-63d50b02ff28-2_422_787_815_340}
   The velocity-time graph illustrates the motion of a particle which accelerates from rest to \(8 \mathrm {~ms} ^ { - 1 }\) in \(x\) seconds and then to \(24 \mathrm {~ms} ^ { - 1 }\) in a further 4 seconds. It then travels at a constant speed for another \(y\) seconds before decelerating to \(12 \mathrm {~ms} ^ { - 1 }\) over the next \(y\) seconds and then to rest in the final 7 seconds of its motion. Given that the total distance travelled by the particle is 496 m ,
2. show that \(2 x + 21 y = 195\). Given also that the average speed of the particle during its motion is \(15 \cdot 5 \mathrm {~ms} ^ { - 1 }\),
3. show that \(x + 2 y = 21\).
4. Hence find the values of \(x\) and \(y\),
5. Write down the acceleration for each section of the motion.

7. Two particles \(P\) and \(Q\), of masses \(2 m\) and \(3 m\) respectively, are connected by a light string. Initially, \(P\) is at rest on a smooth horizontal table. The string passes over a small smooth pulley and \(Q\) rests on a rough plane inclined at an
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angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The coefficient of friction between \(Q\) and the inclined plane is \(\frac { 1 } { 6 }\).
The system is released from rest with \(Q\) at a distance of 0.8 metres above a horizontal floor.

1. Show that the acceleration of \(P\) and \(Q\) is \(\frac { 21 g } { 50 }\), stating a modelling assumption which you must make to ensure that both particles have the same acceleration.
2. Find the speed with which \(Q\) hits the floor. After \(Q\) hits the floor and does not rebound, \(P\) moves a further 0.2 m until it hits the pulley.
3. Find the total time after the system is released before \(P\) hits the pulley.