WJEC Further Unit 3 (Further Unit 3) 2022 June

1. A particle of mass 1.2 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is fixed to a point \(O\) on a smooth horizontal surface. With the string taut, the particle moves on the surface with constant speed \(8 \mathrm {~ms} ^ { - 1 }\) in a horizontal circle with centre \(O\).
   1. Find the angular velocity of the particle about \(O\).
   2. Calculate the tension in the string.
   3. The diagram below shows a woman standing at the end of a diving platform. She is about to dive into the water below.
   The woman has mass 60 kg and she may be modelled as a particle positioned at the end of the platform which is 10 m above the water.
   \includegraphics[max width=\textwidth, alt={}, center]{2954c34b-dd8f-49a2-a379-704022574550-2_750_942_1338_566} When the woman dives, she projects herself from the platform with a speed of \(7.8 \mathrm {~ms} ^ { - 1 }\).
2. Find the kinetic energy of the woman when she leaves the platform.
3. Initially, the situation is modelled ignoring air resistance. By using conservation of energy, show that the model predicts that the woman enters the water with an approximate speed of \(16 \mathrm {~ms} ^ { - 1 }\).
4. Suppose that this model is refined to include air resistance so that the speed with which the woman enters the water is now predicted to be \(13 \mathrm {~ms} ^ { - 1 }\). Determine the amount of energy lost to air resistance according to the refined model.

3. Two spheres \(A\) and \(B\), of equal radii, are moving towards each other on a smooth horizontal surface and collide directly. Sphere \(A\) has mass \(4 m \mathrm {~kg}\) and sphere \(B\) has mass \(3 m \mathrm {~kg}\). Just before the collision, \(A\) has speed \(9 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(3.5 \mathrm {~ms} ^ { - 1 }\). Immediately after the collision, \(A\) has speed \(1.5 \mathrm {~ms} ^ { - 1 }\) in the direction of its original motion.

1. Show that the speed of \(B\) immediately after the collision is \(6.5 \mathrm {~ms} ^ { - 1 }\).
2. Calculate the coefficient of restitution between \(A\) and \(B\).
3. Given that the magnitude of the impulse exerted by \(B\) on \(A\) is 36 Ns , find the value of \(m\).
4. Give a reason why it is not necessary to model the spheres as particles in this question.

4. A particle \(P\) of mass 0.5 kg is in equilibrium under the action of three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\). $$\mathbf { F } _ { 1 } = ( 9 \mathbf { i } + 6 \mathbf { j } - 12 \mathbf { k } ) \mathbf { N } \quad \text { and } \quad \mathbf { F } _ { 2 } = ( 6 \mathbf { i } - 7 \mathbf { j } + 3 \mathbf { k } ) \mathbf { N } .$$

1. Find the force \(\mathbf { F } _ { 3 }\).
2. Forces \(\mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are removed so that \(P\) moves in a straight line \(A B\) under the action of the single force \(\mathbf { F } _ { 1 }\). The points \(A\) and \(B\) have position vectors ( \(2 \mathbf { i } - 9 \mathbf { j } + 7 \mathbf { k }\) )m and \(( 8 \mathbf { i } - 5 \mathbf { j } - \mathbf { k } ) \mathrm { m }\) respectively. The particle \(P\) is initially at rest at \(A\).
   1. Verify that \(\mathbf { F } _ { 1 }\) acts parallel to the vector \(\mathbf { A B }\).
   2. Find the work done by the force \(\mathbf { F } _ { 1 }\) as the particle moves from \(A\) to \(B\).
   3. By using the work-energy principle, find the speed of \(P\) as it reaches \(B\).

5. One end of a light elastic string, of natural length 2.5 m and modulus of elasticity \(30 g \mathrm {~N}\), is fixed to a point \(O\). A particle \(P\), of mass 2 kg , is attached to the other end of the string. Initially, \(P\) is held at rest at the point \(O\). It is then released and allowed to fall under gravity.

1. Show that, while the string is taut, $$v ^ { 2 } = g \left( 5 + 2 x - 6 x ^ { 2 } \right)$$ where \(v \mathrm {~ms} ^ { - 1 }\) denotes the velocity of the particle when the extension in the string is \(x \mathrm {~m}\).
2. Calculate the maximum extension of the string.
3. 1. Find the extension of the string when \(P\) attains its maximum speed.
   2. Hence determine the maximum speed of \(P\).

6. A vehicle of mass 3500 kg is moving up a slope inclined at an angle \(\alpha\) to the horizontal. When the vehicle is travelling at a velocity of \(v \mathrm {~ms} ^ { - 1 }\), the resistance to motion can be modelled by a variable force of magnitude \(40 \nu \mathrm {~N}\).

1. Given that \(\sin \alpha = \frac { 3 } { 49 }\), calculate the power developed by the engine at the instant when the speed of the vehicle is \(25 \mathrm {~ms} ^ { - 1 }\) and its deceleration is \(0.2 \mathrm {~ms} ^ { - 2 }\).
2. When the vehicle's engine is working at a constant rate of 40 kW , the maximum speed that can be maintained up the slope is \(20 \mathrm {~ms} ^ { - 1 }\). Find the value of \(\alpha\). Give your answer in degrees, correct to one significant figure.

7. The diagram below shows a particle \(P\), of mass \(2 \cdot 5 \mathrm {~kg}\), attached by means of two light inextensible strings fixed at points \(A\) and \(B\). Point \(A\) is vertically above point \(B\). \(B P\) makes an angle of \(60 ^ { \circ }\) with the upward vertical and \(A P\) is inclined at an angle \(\theta\) to the downward vertical where \(\cos \theta = 0 \cdot 8\). The particle \(P\) describes a horizontal circle with constant angular speed \(\omega\) radians per second about centre \(C\) with both strings taut.
\includegraphics[max width=\textwidth, alt={}, center]{2954c34b-dd8f-49a2-a379-704022574550-5_823_641_644_705} The tension in the string \(B P\) is \(39 \cdot 2 \mathrm {~N}\).

1. Calculate the tension in the string \(A P\).
2. Given that the length of the string \(A P\) is 1.5 m , find the value of \(\omega\).
3. Calculate the kinetic energy of \(P\).