CAIE M2 (Mechanics 2) 2012 June

1 A particle \(P\) is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. Calculate the distance \(O P\) at the instant 2 s after projection.

2
\includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_448_547_434_799} The diagram shows a uniform object \(A B C\) of weight 3 N in the form of an arc of a circle with centre \(O\) and radius 0.7 m . The angle \(A O C\) is 2 radians. The object rests in equilibrium with \(A\) on a horizontal surface and \(C\) vertically above \(A\). Equilibrium is maintained by a horizontal force of magnitude \(F \mathrm {~N}\) applied at \(C\) in the plane of the object. Calculate \(F\).

3 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\), and moves in a straight line on a smooth horizontal surface. A force of magnitude \(0.4 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Given that \(P\) comes to instantaneous rest when \(x = 2.5\), find the initial kinetic energy of \(P\).
2. Find the value of \(x\) on the first occasion when the speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4
\includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_170_616_1649_767} A small sphere \(S\) of mass \(m \mathrm {~kg}\) is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(S\) moves with constant speed in a horizontal circle of radius 0.4 m and is in contact with both the plane base and the curved surface of the cylinder (see diagram).

1. Given that the horizontal and vertical forces exerted on \(S\) by the cylinder have equal magnitudes, calculate the speed of \(S\).
   \(S\) is now attached to the centre of the base of the cylinder by a horizontal light elastic string of natural length 0.25 m and modulus of elasticity 13 N . The sphere \(S\) is set in motion and moves in a horizontal circle with constant angular speed \(\omega \mathrm { rads } ^ { - 1 }\) and is in contact with both the plane base and the curved surface of the cylinder.
2. It is given that the magnitudes of the horizontal and vertical forces exerted on \(S\) by the cylinder are equal if \(\omega = 8\). Calculate \(m\).
3. For the value of \(m\) found in part (ii), find the least possible value of \(\omega\) for the motion.

5 A light elastic string has natural length 3 m and modulus of elasticity 45 N . A particle \(P\) of mass 0.6 kg is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie on a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is 4 m , and \(A\) is higher than \(B\).

1. Calculate the distance \(A P\) when \(P\) rests on the slope in equilibrium.
   \(P\) is released from rest at the point between \(A\) and \(B\) where \(A P = 2.5 \mathrm {~m}\).
2. Find the maximum speed of \(P\).
3. Show that \(P\) is at rest when \(A P = 1.6 \mathrm {~m}\).

6
\includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-3_341_791_886_678} A uniform lamina \(A B C D E\) consists of a rectangle \(B C D E\) and an isosceles triangle \(A B E\) joined along their common edge \(B E\). For the triangle, \(A B = A E , B E = a \mathrm {~m}\) and the perpendicular height is \(h \mathrm {~m}\). For the rectangle, \(B C = D E = 0.5 \mathrm {~m}\) and \(C D = B E = a \mathrm {~m}\) (see diagram).

1. Show that the distance in metres of the centre of mass of the lamina from \(B E\) towards \(C D\) is $$\frac { 3 - 4 h ^ { 2 } } { 12 + 12 h }$$ The lamina is freely suspended at \(E\) and hangs in equilibrium.
2. Given that \(D E\) is horizontal, calculate \(h\).
3. Given instead that \(h = 0.5\) and \(A E\) is horizontal, calculate \(a\).

7 The equation of the trajectory of a projectile is \(y = 0.6 x - 0.017 x ^ { 2 }\), referred to horizontal and vertically upward axes through the point of projection.

1. Find the angle of projection of the projectile, and show that the initial speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the speed and direction of motion of the projectile when it is at a height of 5.2 m above the level of the point of projection for the second time.