Edexcel M1 (Mechanics 1)

Briefly define the following terms used in modelling in Mechanics:

1. lamina,
2. uniform rod,
3. smooth surface,
4. particle.

[4 marks]

Two forces \(\mathbf{F}\) and \(\mathbf{G}\) are given by \(\mathbf{F} = (6\mathbf{i} - 5\mathbf{j})\) N, \(\mathbf{G} = (3\mathbf{i} + 17\mathbf{j})\) 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.

1. Find the magnitude of \(\mathbf{R}\), the resultant of \(\mathbf{F}\) and \(\mathbf{G}\). [3 marks]
2. Find the angle between the direction of \(\mathbf{R}\) and the positive \(x\)-axis. [2 marks]

\(\mathbf{R}\) acts through the point \(P(-4, 3)\). \(O\) is the origin \((0, 0)\).

1. Use the fact that \(OP\) 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. [3 marks]

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 the string makes an angle of 35° with the vertical as shown, the case is on the point of moving. \includegraphics{figure_3}

1. Find the coefficient of friction between the case and the plane. [5 marks]

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 ms\(^{-1}\) after 4 seconds.

1. Find the magnitude of the new force. [5 marks]
2. State any modelling assumptions you have made about the case and the string. [2 marks]

A uniform yoke \(AB\), of mass 4 kg and length 4\(a\) m, rests on the shoulders \(S\) and \(T\) of two oxen. \(AS = TB = a\) m. A bucket of mass \(x\) kg is suspended from \(A\). \includegraphics{figure_4}

1. Show that the vertical force on the yoke at \(T\) has magnitude \((2 - \frac{1}{4}x)g\) N and find, in terms of \(x\) and \(g\), the vertical force on the yoke at \(S\). [7 marks]
2. If the ratio of these vertical forces is \(5 : 1\), find the value of \(x\). [3 marks]
3. Find the maximum value of \(x\) for which the yoke will remain horizontal. [2 marks]

Two small smooth spheres \(A\) and \(B\), of equal radius but masses \(m\) 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 ms\(^{-1}\) and \(B\) has speed 3 ms\(^{-1}\). In the collision, the impulse exerted by \(A\) on \(B\) has magnitude \(7km\) Ns.

1. Find the speed of \(B\) after the impact. [3 marks]
2. Show that the speed of \(A\) immediately after the collision is \((7k - 5)\) ms\(^{-1}\) and deduce that the direction of \(A\)'s motion is reversed. [5 marks]

\(B\) is now given a further impulse of magnitude \(mu\) Ns, as a result of which a second collision between it and \(A\) occurs.

1. Show that \(u > k(7k - 1)\). [4 marks]

The velocity-time graph illustrates the motion of a particle which accelerates from rest to 8 ms\(^{-1}\) in \(x\) seconds and then to 24 ms\(^{-1}\) in a further 4 seconds. It then travels at a constant speed for another \(y\) seconds before decelerating to 12 ms\(^{-1}\) over the next \(y\) seconds and then to rest in the final 7 seconds of its motion. \includegraphics{figure_6} Given that the total distance travelled by the particle is 496 m,

1. show that \(2x + 21y = 195\). [4 marks]

Given also that the average speed of the particle during its motion is 15.5 ms\(^{-1}\),

1. show that \(x + 2y = 21\). [3 marks]
2. Hence find the values of \(x\) and \(y\). [3 marks]
3. Write down the acceleration for each section of the motion. [3 marks]

Two particles \(P\) and \(Q\), of masses \(2m\) and \(3m\) 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 angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{4}{3}\). The coefficient of friction between \(Q\) and the inclined plane is \(\frac{1}{6}\). \includegraphics{figure_7} 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 \(\frac{21g}{50}\), stating a modelling assumption which you must make to ensure that both particles have the same acceleration. [7 marks]
2. Find the speed with which \(Q\) hits the floor. [2 marks]

After \(Q\) hits the floor and does not rebound, \(P\) moves a further 0.2 m until it hits the pulley.

1. Find the total time after the system is released before \(P\) hits the pulley. [5 marks]