Edexcel M1 (Mechanics 1)

1. Two forces, both of magnitude 5 N , act on a particle in the directions with bearings \(000 ^ { \circ }\) and \(070 ^ { \circ }\), as shown. Calculate
   1. the magnitude of the resultant force on the particle,
   2. the bearing on which this resultant force acts.
   3. A uniform plank \(X Y\) has length 7 m and mass 2 kg . It is placed with the portion \(Z Y\) in contact with a
      \includegraphics[max width=\textwidth, alt={}, center]{38e355b0-9d75-40ad-b450-bd74c5135c7f-1_149_616_843_1334}
      horizontal surface, where \(Z Y = 2.8 \mathrm {~m}\). To prevent the
      \includegraphics[max width=\textwidth, alt={}, center]{38e355b0-9d75-40ad-b450-bd74c5135c7f-1_207_253_404_1505}
   \section*{MECHANICS 1 (A) TEST PAPER 5 Page 2}

1. \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. The point \(A\) has position vector \(6 \mathbf { j } \mathrm {~m}\) relative to an origin \(O\). At time \(t = 0\) a particle \(P\) starts from \(O\) and moves with constant velocity ( \(5 \mathbf { i } + 2 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\). At the same instant a particle \(Q\) starts from \(A\) and moves with constant velocity \(4 \mathrm { ims } ^ { - 1 }\).
   1. Write down the position vectors of \(P\) and of \(Q\) at time \(t\) seconds.
   2. Show that the distance \(d \mathrm {~m}\) between \(P\) and \(Q\) at time \(t\) seconds is such that
   $$d ^ { 2 } = 5 t ^ { 2 } - 24 t + 36 .$$
2. Find the value of \(t\) for which \(d ^ { 2 }\) is a minimum.
3. Hence find the minimum distance between \(P\) and \(Q\), and state the position vector of each particle when they are closest together.

6. \(A , B\) and \(C\) are three small spheres of equal radii and masses \(2 m , m\) and \(5 m\) respectively. They are placed in a straight line on a smooth horizontal surface. \(A\) is projected with speed \(6 \mathrm {~ms} ^ { - 1 }\) towards \(B\), which is at rest. When \(A\) hits \(B\) it exerts an impulse of magnitude 8 m Ns on \(B\).

1. Find the speed with which \(B\) starts to move.
2. Show that the speed of \(A\) after it collides with \(B\) is \(2 \mathrm {~ms} ^ { - 1 }\). After travelling \(3 \mathrm {~m} , B\) hits \(C\), which is then travelling towards \(B\) at \(2 \cdot 2 \mathrm {~ms} ^ { - 1 } . C\) is brought to rest by this impact.
3. Show that the direction of \(B\) 's motion is reversed and find its new speed.
4. Find how far \(B\) now travels before it collides with \(A\) again.
5. State a modelling assumption that you have made about the spheres.

7. A particle \(P\), of mass \(m\), is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal as shown. A light string is attached to \(P\) and makes an angle of \(30 ^ { \circ }\) with the plane. When the tension in this string has magnitude \(k m g , P\) is
\includegraphics[max width=\textwidth, alt={}, center]{38e355b0-9d75-40ad-b450-bd74c5135c7f-2_268_474_1759_1407}
just on the point of moving up the plane.

1. Show that \(\mu\), the coefficient of friction between \(P\) and the plane, is \(\frac { k \sqrt { } 3 - 1 } { \sqrt { } 3 - k }\).
2. Given further that \(k = \frac { 3 \sqrt { } 3 } { 7 }\), deduce that \(\mu = \frac { \sqrt { } 3 } { 6 }\). The string is now removed.
3. Determine whether \(P\) will move down the plane and, if it does, find its acceleration.
4. Give a reason why the way in which \(P\) is shown in the diagram might not be consistent with the modelling assumptions that have been made.