Edexcel M3 (Mechanics 3)

1. A particle of mass \(m \mathrm {~kg}\) moves in a horizontal straight line. Its initial speed is \(u \mathrm {~ms} ^ { - 1 }\) and the only force acting on it is a variable resistance of magnitude \(m k v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the particle after \(t\) seconds and \(k\) is a constant.
   Show that \(v = u e ^ { - k t }\).
2. A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal circle at one end of a light inextensible string of length 40 cm , as shown. The other end of the string is attached to a fixed point \(O\).
   The angular velocity of \(P\) is \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\).
   If the angle \(\theta\) which the string makes with the vertical must not
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   (7 marks)
   exceed \(60 ^ { \circ }\), calculate the greatest possible value of \(\omega\).
3. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity \(\frac { m g } { 2 } \mathrm {~N}\). The other end of the string is attached to a fixed point \(O\) and \(P\) hangs vertically below \(O\).
   1. Find the stretched length of the string when \(P\) rests in equilibrium.
   2. Find the elastic potential energy stored in the string in the equilibrium position. \(P\), which is still attached to the string, is now held at rest at \(O\) and then lowered gently into its equilibrium position.
   3. Find the work done by the weight of the particle as it moves from \(O\) to the equilibrium position.
   4. Explain the discrepancy between your answers to parts (b) and (c).
   5. A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to two light elastic strings, each of natural length \(l \mathrm {~m}\) and modulus of elasticity 3 mg N . The other ends of the strings are attached to the fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 2 l \mathrm {~m}\). If \(P\) rests in equilibrium vertically below the mid-point of \(A B\), with each string making an angle \(\theta\) with the vertical, show that
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   $$\cot \theta - \cos \theta = \frac { 1 } { 6 } .$$ \section*{MECHANICS 3 (A) TEST PAPER 2 Page 2}

1. A small bead \(P\), of mass \(m \mathrm {~kg}\), can slide on a smooth circular ring, with centre \(O\) and radius \(r \mathrm {~m}\), which is fixed in a vertical plane. \(P\) is projected from the lowest point \(L\) of the ring with speed \(\sqrt { } ( 3 g r ) \mathrm { ms } ^ { - 1 }\). When \(P\) has reached a position such that \(O P\) makes an angle \(\theta\) with the downward vertical, as shown, its speed is \(v \mathrm {~ms} ^ { - 1 }\).
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   1. Show that \(v ^ { 2 } = g r ( 1 + 2 \cos \theta )\).
   2. Show that the magnitude of the reaction \(R N\) of the ring on the bead is given by
   $$R = m g ( 1 + 3 \cos \theta ) .$$
2. Find the values of \(\cos \theta\) when
   1. \(P\) is instantaneously at rest, (ii) the reaction \(R\) is instantaneously zero.
3. Hence show that the ratio of the heights of \(P\) above \(L\) in cases (i) and (ii) is \(9 : 8\).

6. A light elastic string, of natural length 0.8 m , has one end fastened to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.5 kg . When \(P\) hangs in equilibrium, the length of the string is 1.5 m .

1. Calculate the modulus of elasticity of the string.
   \(P\) is displaced to a point 0.5 m vertically below its equilibrium position and released from rest.
2. Show that the subsequent motion of \(P\) is simple harmonic, with period 1.68 s .
3. Calculate the maximum speed of \(P\) during its motion.
4. Show that the time taken for \(P\) to first reach a distance 0.25 m from the point of release is 0.28 s , to 2 significant figures.

7. (a) Show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 r } { 8 }\) from the centre \(O\) of the plane face. The figure shows the vertical cross-section of a rough solid hemisphere at rest on a rough inclined plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 10 }\).
(b) Indicate on a copy of the figure the three forces acting on the hemisphere, clearly stating what they are, and paying
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(c) Given that the plane face containing the diameter \(A B\) makes an angle \(\alpha\) with the vertical, show that \(\cos \alpha = \frac { 4 } { 5 }\).