OCR Further Mechanics (Further Mechanics) 2021 June

1 A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32 N . The rope has natural length 4 m and modulus of elasticity 470 N . By considering energy, determine the total distance she falls before first coming to instantaneous rest.

2 One end of a light inextensible string of length 0.75 m is attached to a particle \(A\) of mass 2.8 kg . The other end of the string is attached to a fixed point \(O . A\) is projected horizontally with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 0.75 m vertically above \(O\) (see Fig. 2). When \(O A\) makes an angle \(\theta\) with the upward vertical the speed of \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(\xrightarrow [ A \text { a } ] { 6 \mathrm {~m} \mathrm {~s} ^ { - 1 } }\) Fig. 2

1. Show that \(v ^ { 2 } = 50.7 - 14.7 \cos \theta\).
2. Given that the string breaks when the tension in it reaches 200 N , find the angle that \(O A\) turns through between the instant that \(A\) is projected and the instant that the string breaks.

3 The resistive force, \(F\), on a sphere falling through a viscous fluid is thought to depend on the radius of the sphere, \(r\), the velocity of the sphere, \(v\), and the viscosity of the fluid, \(\eta\). You are given that \(\eta\) is measured in \(\mathrm { Nm } ^ { - 2 } \mathrm {~s}\).

1. By considering its units, find the dimensions of viscosity. A model of the resistive force suggests the following relationship: \(F = 6 \pi \eta ^ { \alpha } r ^ { \beta } v ^ { \gamma }\).
2. Explain whether or not it is possible to use dimensional analysis to verify that the constant \(6 \pi\) is correct.
3. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\). A sphere of radius \(r\) and mass \(m\) falls vertically from rest through the fluid. After a time \(t\) its velocity is \(v\).
4. By setting up and solving a differential equation, show that \(\mathrm { e } ^ { - k t } = \frac { g - k v } { g }\) where \(k = \frac { 6 \pi \eta r } { m }\). As the time increases, the velocity of the sphere tends towards a limit called the terminal velocity.
5. Find, in terms of \(g\) and \(k\), the terminal velocity of the sphere. In a sequence of experiments the sphere is allowed to fall through fluids of different viscosity, ranging from small to very large, with all other conditions being constant. The terminal velocity of the sphere through each fluid is measured.
6. Describe how, according to the model, the terminal velocity of the sphere changes as the viscosity of the fluid through which it falls increases.

4 Fig. 4.1 shows a uniform lamina in the shape of a sector of a circle of radius \(r\) and angle \(2 \theta\) where \(\theta\) is in radians. The sector consists of a triangle \(O A B\) and a segment bounded by the chord \(A B\). \begin{figure}[h] \end{figure}

1. Explain why the centre of mass of the segment lies on the radius through the midpoint of \(A B\).
2. Show that the distance of the centre of mass of the segment from \(O\) is \(\frac { 2 r \sin ^ { 3 } \theta } { 3 ( \theta - \sin \theta \cos \theta ) }\). A uniform circular lamina of radius 5 units is placed with its centre at the origin, \(O\), of an \(x - y\) coordinate system. A component for a machine is made by removing and discarding a segment from the lamina. The radius of the circle from which the segment is formed is 3 units and the centre of this circle is \(O\). The centre of the straight edge of the segment has coordinates \(( 0,2 )\) and this edge is perpendicular to the \(y\)-axis (see Fig. 4.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8859baf3-f8e8-4fbf-b54f-34f550b02c26-03_748_743_1594_251} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
3. Find the \(y\)-coordinate of the centre of mass of the component, giving your answer correct to 3 significant figures. \(C\) is the point on the component with coordinates \(( 0,5 )\). The component is now placed horizontally and supported only at \(O\). A particle of mass \(m \mathrm {~kg}\) is placed on the component at \(C\) and the component and particle are in equilibrium.
4. Find the mass of the component in terms of \(m\). Total Marks for Question Set 3: 40