SPS SPS FM Mechanics (SPS FM Mechanics) 2021 May

2.
\includegraphics[max width=\textwidth, alt={}, center]{ba21d750-a058-43c7-b602-2bafe545b94a-06_662_540_376_742} 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 \(\theta\).
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[0pt] [Question 2 Continued]
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3. 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. 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. The car is now driven at a constant speed of \(30 \mathrm {~ms} ^ { - 1 }\) 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.
      [0pt] [Question 3 Continued]
      [0pt] [Question 3 Continued]
      [0pt] [Question 3 Continued]

4.
\includegraphics[max width=\textwidth, alt={}, center]{ba21d750-a058-43c7-b602-2bafe545b94a-14_357_840_445_552} 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 \(2.8 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving with speed \(1 \mathrm {~ms} ^ { - 1 }\) 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 \(B\),
2. the angle turned through by the direction of motion of B as a result of the collision.
   [0pt] [Question 4 Continued]
   [0pt] [Question 4 Continued]
   [0pt] [Question 4 Continued] \section*{5.} 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 .
3. Find the tension in the string.
4. Find the speed of \(P\).
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   [0pt] [Question 5 Continued] \section*{6.} The figure below shows the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(y = 8\), and the curve \(y = ( x - 2 ) ^ { 3 }\) for \(0 \leq y \leq 8\).
   \includegraphics[max width=\textwidth, alt={}, center]{ba21d750-a058-43c7-b602-2bafe545b94a-22_595_643_523_680} Find the coordinates of the centre of mass of a uniform lamina occupying this region. No marks will be deducted for using the numerical integration function of your calculator for this question.
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   [0pt] [Question 6 Continued]
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