Edexcel M3 (Mechanics 3)

1. A cyclist travels on a banked track inclined at \(8 ^ { \circ }\) to the horizontal. He moves in a horizontal circle of radius 10 m at a constant speed of \(v \mathrm {~ms} ^ { - 1 }\). If there is no sideways frictional force on the cycle, calculate the value of \(v\).
2. The figure shows a particle \(P\), of mass 0.8 kg , attached to the ends of two light elastic strings. \(A P\) has natural length 20 cm and modulus of elasticity \(\lambda \mathrm { N } . B P\) has natural length 20 cm and modulus of of elasticity \(\mu \mathrm { N } . A\) and \(B\) are fixed to points on the same horizontal
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   level so that \(A B = 50 \mathrm {~cm}\). When \(P\) is suspended in equilibrium, \(A P =\) 30 cm and \(B P = 40 \mathrm {~cm}\). Calculate the values of \(\lambda\) and \(\mu\).
3. Suraiya, whose mass is \(m \mathrm {~kg}\), takes a running jump into a swimming pool so that she begins to swim in a straight line with speed \(0.2 \mathrm {~ms} ^ { - 1 }\). She continues to move in the same straight line, the only force acting on her being a resistance of magnitude \(m v ^ { 2 } \sin \left( \frac { t } { 100 } \right) \mathrm { N }\), where \(v \mathrm {~ms} ^ { - 1 }\) is her speed at time \(t\) seconds after entering the pool and \(0 \leq t \leq 50 \pi\).
   1. Find an expression for \(v\) in terms of \(t\).
   2. Calculate her greatest and least speeds during her motion.
   3. A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation \(y = 1 + \cos x\), as shown.
   4. Show by integration that the centre of mass of the lamina is at a distance \(\frac { \pi ^ { 2 } - 4 } { 2 \pi }\) from the \(y\)-axis.
      (9 marks)
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   Given that the centre of mass is at a distance 0.75 units from the \(x\)-axis, and that \(P\) is the point \(( 0,2 )\) and \(O\) is the origin \(( 0,0 )\),
4. find, to the nearest degree, the angle between the line \(O P\) and the vertical when the lamina is freely suspended from \(P\).

5. A particle \(P\), of mass 0.5 kg , rests on the surface of a rough horizontal table. The coefficient of friction between \(P\) and the table is \(0.5 . P\) is connected to a particle \(Q\), of mass 0.2 kg , by a light inextensible string passing through
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   a small smooth hole at a point \(O\) on the table, such that the distance \(O Q\) is \(0.4 \mathrm {~m} . Q\) moves in a horizontal circle while \(P\) remains in limiting equilibrium.
   1. Calculate the angle \(\theta\) which \(O Q\) makes with the vertical.
   2. Show that the speed of \(Q\) is \(1.33 \mathrm {~ms} ^ { - 1 }\).
   The motion is altered so that \(Q\) hangs at rest below \(O\) and \(P\) moves in a horizontal circle on the table with speed \(0.84 \mathrm {~ms} ^ { - 1 }\), at a constant distance \(r \mathrm {~m}\) from \(O\) but tending to slip away from \(O\).
2. Find the value of \(r\).

6. The figure shows a swing consisting of two identical vertical light springs attached symmetrically to a light horizontal cross-bar and supported from a strong fixed horizontal beam. When a mass of 24 kg is placed at the mid-
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point of the cross-bar, both springs extend by 30 cm to the position \(A\), as shown. Each spring has natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\).

1. Show that \(\lambda = 392 l\). The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm .
2. Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level \(A\).
3. Calculate the number of oscillations made per second in this motion.
4. Find the maximum acceleration which the mass experiences during the motion.

7. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to points \(C\) and \(D\) on the same horizontal level by means of two light inextensible strings \(C P\) and \(D P\), both of length \(40 \mathrm {~cm} . P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) so as to move in a vertical circle in a plane perpendicular to \(C D\), so that angle \(P C D =\) angle \(P D C = \theta\) throughout the motion.
\includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-2_335_405_1775_1572} If \(u\) is just large enough for the strings to remain taut as \(P\) describes this circular path,

1. show that \(u ^ { 2 } = 2 g \sin \theta\). The string \(D P\) breaks when \(P\) is at its lowest point. \(P\) then immediately starts to move in a horizontal circle on the end of the string \(C P\).
2. Prove that \(\tan \theta = \frac { 1 } { 5 } \sqrt { 5 }\).