Edexcel M1 (Mechanics 1)

Two forces, both of magnitude 5 N, act on a particle in the directions with bearings 000° and 070°, as shown. \includegraphics{figure_1} Calculate

1. the magnitude of the resultant force on the particle, [3 marks]
2. the bearing on which this resultant force acts. [2 marks]

A uniform plank \(XY\) has length 7 m and mass 2 kg. It is placed with the portion \(ZY\) in contact with a horizontal surface, where \(ZY = 2.8\) m. To prevent the plank from toppling, a stone is placed on the plank at \(Y\). \includegraphics{figure_2}

1. Find the smallest possible mass of the stone. [4 marks]
2. State, with a reason, whether your answer to part (a) would be greater or smaller if a shorter portion of the plank were in contact with the surface. [2 marks]

A car, of mass 1800 kg, pulls a trailer of mass 350 kg along a straight horizontal road. When the car is accelerating at \(0.2\) ms\(^{-2}\), the resistances to the motion of the car and trailer have magnitudes 300 N and 100 N respectively. Find, at this time,

1. the driving force produced by the engine of the car, [3 marks]
2. the tension in the tow-bar between the car and the trailer. [4 marks]

A train starts from rest at a station \(S\) and accelerates at a constant rate for \(2x\) seconds to a speed of \(5x\) ms\(^{-1}\). It maintains this speed until 126 seconds after it left \(S\) and then decelerates at a constant rate until it comes to rest at another station \(T\), \(20x\) seconds after it left \(S\).

1. Sketch a velocity-time graph for this journey. [4 marks]

Given that the distance between \(S\) and \(T\) is \(5.4\) km,

1. show that \(x^2 + 7x = 120\). [4 marks]
2. Find the value of \(x\). [3 marks]

\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. The point \(A\) has position vector \(6\mathbf{j}\) m relative to an origin \(O\). At time \(t = 0\) a particle \(P\) starts from \(O\) and moves with constant velocity \((5\mathbf{i} + 2\mathbf{j})\) ms\(^{-1}\). At the same instant a particle \(Q\) starts from \(A\) and moves with constant velocity \(4\mathbf{i}\) ms\(^{-1}\).

1. Write down the position vectors of \(P\) and of \(Q\) at time \(t\) seconds. [3 marks]
2. Show that the distance \(d\) m between \(P\) and \(Q\) at time \(t\) seconds is such that $$d^2 = 5t^2 - 24t + 36.$$ [5 marks]
3. Find the value of \(t\) for which \(d^2\) is a minimum. [3 marks]
4. Hence find the minimum distance between \(P\) and \(Q\), and state the position vector of each particle when they are closest together. [4 marks]

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

1. Find the speed with which \(B\) starts to move. [2 marks]
2. Show that the speed of \(A\) after it collides with \(B\) is 2 ms\(^{-1}\). [3 marks]

After travelling 3 m, \(B\) hits \(C\), which is then travelling towards \(B\) at \(2.2\) ms\(^{-1}\). \(C\) is brought to rest by this impact.

1. Show that the direction of \(B\)'s motion is reversed and find its new speed. [3 marks]
2. Find how far \(B\) now travels before it collides with \(A\) again. [6 marks]
3. State a modelling assumption that you have made about the spheres. [1 mark]

A particle \(P\), of mass \(m\), is in contact with a rough plane inclined at 30° to the horizontal as shown. A light string is attached to \(P\) and makes an angle of 30° with the plane. When the tension in this string has magnitude \(kmg\), \(P\) is just on the point of moving up the plane. \includegraphics{figure_7}

1. Show that \(\mu\), the coefficient of friction between \(P\) and the plane, is \(\frac{k\sqrt{3} - 1}{\sqrt{3} - k}\). [7 marks]
2. Given further that \(k = \frac{3\sqrt{3}}{7}\), deduce that \(\mu = \frac{\sqrt{3}}{6}\). [3 marks]

The string is now removed.

1. Determine whether \(P\) will move down the plane and, if it does, find its acceleration. [5 marks]
2. 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. [1 mark]