AQA M2 (Mechanics 2) 2015 June

1 A particle, of mass 4 kg , moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane, perpendicular to each other. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \cos 2 t \mathbf { i } + 3 \sin t \mathbf { j }$$

1. 1. Find an expression for the force, \(\mathbf { F }\), acting on the particle at time \(t\) seconds.
   2. Find the magnitude of \(\mathbf { F }\) when \(t = \pi\).
2. When \(t = 0\), the particle is at the point with position vector \(( 2 \mathbf { i } - 14 \mathbf { j } )\) metres. Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\) seconds.
   [0pt] [5 marks]
   \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-02_1346_1717_1361_150}

2 A uniform rod \(A B\), of mass 4 kg and length 6 metres, has three masses attached to it. A 3 kg mass is attached at the end \(A\) and a 5 kg mass is attached at the end \(B\). An 8 kg mass is attached at a point \(C\) on the rod. Find the distance \(A C\) if the centre of mass of the system is 4.3 m from point \(A\).
[0pt] [4 marks]

3 A diagram shows a children's slide, \(P Q R\).
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_352_640_338_699} Simon, a child of mass 32 kg , uses the slide, starting from rest at \(P\). The curved section of the slide, \(P Q\), is one sixth of a circle of radius 4 metres so that the child is travelling horizontally at point \(Q\). The centre of this circle is at point \(O\), which is vertically above point \(Q\). The section \(Q R\) is horizontal and of length 5 metres. Assume that air resistance may be ignored.

1. Assume that the two sections of the slide, \(P Q\) and \(Q R\), are both smooth.
   1. Find the kinetic energy of Simon when he reaches the point \(R\).
   2. Hence find the speed of Simon when he reaches the point \(R\).
2. In fact, the section \(Q R\) is rough. Assume that the section \(P Q\) is smooth.
   Find the coefficient of friction between Simon and the section \(Q R\) if Simon comes to rest at the point \(R\).
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_923_1707_1784_153}

4 A particle, \(P\), of mass 5 kg is attached to two light inextensible strings, \(A P\) and \(B P\). The other ends of the strings are attached to the fixed points \(A\) and \(B\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in a horizontal circle of radius 0.6 metres with centre \(B\). The string \(A P\) is inclined at \(20 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut when the particle is moving.
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-08_835_568_568_719}

1. Find the tension in the string \(A P\).
2. The speed of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the tension, \(T _ { B P }\), in the string \(B P\) is given by $$T _ { B P } = \frac { 25 } { 3 } v ^ { 2 } - 5 g \tan 20 ^ { \circ }$$
3. Find \(v\) when the tensions in the two strings are equal.

5 An item of clothing is placed inside a washing machine. The drum of the washing machine has radius 30 cm and rotates, about a fixed horizontal axis, at a constant angular speed of 900 revolutions per minute. Model the item of clothing as a particle of mass 0.8 kg and assume that the clothing travels in a vertical circle with constant angular speed. Find the minimum magnitude of the normal reaction force exerted by the drum on the clothing and find the maximum magnitude of the normal reaction force exerted by the drum on the clothing.
[0pt] [6 marks]

6 A van, of mass 1400 kg , is accelerating at a constant rate of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it travels up a slope inclined at an angle \(\theta\) to the horizontal. The van experiences total resistance forces of 4000 N .
When the van is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power output of the van's engine is 91.1 kW . Find \(\theta\).
[0pt] [9 marks]

7 A parachutist, of mass 72 kg , is falling vertically. He opens his parachute at time \(t = 0\) when his speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then experiences an air resistance force of magnitude \(240 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is his speed at time \(t\) seconds.

1. When \(t > 0\), show that \(- \frac { 3 } { 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = v - 2.94\).
2. Find \(v\) in terms of \(t\).
3. Sketch a graph to show how, for \(t \geqslant 0\), the parachutist's speed varies with time.
   [0pt] [2 marks]

8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen \(x \mathrm {~m}\), her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
2. Why is the expression found in part (a) not true when \(x\) takes values less than 26?
3. Find the maximum value of \(x\).
4. 1. Find the distance fallen by Carol when her speed is a maximum.
   2. Hence find Carol's maximum speed.

9 A uniform rod, \(P Q\), of length \(2 a\), rests with one end, \(P\), on rough horizontal ground and a point \(T\) resting on a rough fixed prism of semicircular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both \(P\) and \(T\) is \(\mu\).
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475} The rod is on the point of slipping when it is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Find the value of \(\mu\).
[0pt] [8 marks]
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}