SPS SPS FM Mechanics (SPS FM Mechanics) 2022 January

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 32N. The rope has natural length 4 m and modulus of elasticity of 470 N. By considering energy, determine the total distance she falls before first coming to instantaneous rest. [6]

\includegraphics{figure_2} A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac{1}{3}\). The cone is freely suspended by a string attached at a point A on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta^0\) with the upward vertical, as shown in the diagram above. Find, to one decimal place, the value of \(\theta\). [5]

A car of mass 800 kg is driven with its engine generating a power of 15 kW.

1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds. [2]
2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude of 150 N. Given that \(\sin \theta = \frac{1}{20}\), find the speed of the car. [3]
3. The car is now driven at a constant speed of 30 ms\(^{-1}\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming the resistance to motion of the car is three times the resistance to motion of the trailer. Find:
   1. the resistance to motion of the car,
   2. the magnitude of the tension in the towbar
   [4]

\includegraphics{figure_4} Two uniform smooth spheres A and B of equal radius are moving on a horizontal surface when they collide. A has mass 0.1 kg and B has mass 0.4 kg. Immediately before the collision A is moving with speed 2.8 ms\(^{-1}\) along the line of centres, and B is moving with speed 1 ms\(^{-1}\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision A is stationary. Find:

1. the coefficient of restitution between A and B, [5]
2. the angle turned through by the direction of motion of B as a result of the collision. [4]

A right circular cone C of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of C. The other end of the string is attached to a particle P of mass 2.5 kg. P moves in a horizontal circle with constant speed and in contact with the smooth curved surface of C. The extension of the string is 1.5 m.

1. Find the tension in the string. [2]
2. Find the speed of P. [7]

A uniform rod, PQ, of length \(2a\), rests with one end, P, on rough horizontal ground and a point T resting on a rough fixed prism of semi-circular 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{figure_6} The rod is on the point of slipping when it is inclined at an angle of 30\(^0\) to the horizontal. Find the value of \(\mu\). [8]

The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium ABCD with AB and CD perpendicular to AD. The lengths of AB and AD are each 5 cm and the length of CD is \((a + 5)\) cm. \includegraphics{figure_7}

1. Show the distance of the centre of mass of the prism from AD is $$\frac{a^2 + 15a + 75}{3(a + 10)} \text{ cm.}$$ [5]

The prism is placed with the face containing AB in contact with a horizontal surface.

1. Find the greatest value of \(a\) for which the prism does not topple. [3]

The prism is now placed on an inclined plane which makes an angle \(\theta^o\) with the horizontal. AB lies along a line of greatest slope with B higher than A.

1. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the great value of \(\theta\) for which the prism does not topple. [6]