CAIE M2 (Mechanics 2) 2011 June

1 A particle is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal from a point on horizontal ground. Calculate the time taken for the particle to hit the ground.

2
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\(A O B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.6 m (see diagram).

1. Calculate the distance of the centre of mass of the lamina from \(A\). The lamina is freely suspended at \(A\) and hangs in equilibrium.
2. Find the angle between the vertical and the side \(A O\) of the lamina.

3
\includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_247_839_1375_653} A light elastic string of natural length 1.2 m and modulus of elasticity 24 N is attached to fixed points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 1.2 \mathrm {~m}\). A particle \(P\) is attached to the mid-point of the string. \(P\) is projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the surface in a direction perpendicular to \(A B\) (see diagram). \(P\) comes to instantaneous rest at a distance 0.25 m from \(A B\).

1. Show that the mass of \(P\) is 0.8 kg .
2. Calculate the greatest deceleration of \(P\).

4 A particle \(P\) starts from rest at a point \(O\) and travels in a straight line. The acceleration of \(P\) is \(( 15 - 6 x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Find the value of \(x\) for which \(P\) reaches its maximum velocity, and calculate this maximum velocity.
2. Calculate the acceleration of \(P\) when it is at instantaneous rest and \(x > 0\).

5
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\(A B C\) is a uniform triangular lamina of weight 19 N , with \(A B = 0.22 \mathrm {~m}\) and \(A C = B C = 0.61 \mathrm {~m}\). The plane of the lamina is vertical. \(A\) rests on a rough horizontal surface, and \(A B\) is vertical. The equilibrium of the lamina is maintained by a light elastic string of natural length 0.7 m which passes over a small smooth peg \(P\) and is attached to \(B\) and \(C\). The portion of the string attached to \(B\) is horizontal, and the portion of the string attached to \(C\) is vertical (see diagram).

1. Show that the tension in the string is 10 N .
2. Calculate the modulus of elasticity of the string.
3. Find the magnitude and direction of the force exerted by the surface on the lamina at \(A\).

6 A particle \(P\) is projected from a point \(O\) on horizontal ground. 0.4 s after the instant of projection, \(P\) is 5 m above the ground and a horizontal distance of 12 m from \(O\).

1. Calculate the initial speed and the angle of projection of \(P\).
2. Find the direction of motion of the particle 0.4 s after the instant of projection.

7
\includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-4_713_933_258_605} A narrow groove is cut along a diameter in the surface of a horizontal disc with centre \(O\). Particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, lie in the groove, and the coefficient of friction between each of the particles and the groove is \(\mu\). The particles are attached to opposite ends of a light inextensible string of length 1 m . The disc rotates with angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis passing through \(O\) and the particles move in horizontal circles (see diagram).

1. Given that \(\mu = 0.36\) and that both \(P\) and \(Q\) move in the same horizontal circle of radius 0.5 m , calculate the greatest possible value of \(\omega\) and the corresponding tension in the string.
2. Given instead that \(\mu = 0\) and that the tension in the string is 0.48 N , calculate
   (a) the radius of the circle in which \(P\) moves and the radius of the circle in which \(Q\) moves,
   (b) the speeds of the particles.