CAIE M2 (Mechanics 2) 2019 June

1
\includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-03_231_970_258_591} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.8 m . The fixed point \(O\) is 0.15 m vertically below \(A\). The particle \(P\) moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with centre \(O\) (see diagram).

1. Show that the tension in the string is 16 N .
2. Find the value of \(v\).

2 A particle is projected with speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. At the instant 4 s after projection the speed of the particle is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its direction of motion is \(30 ^ { \circ }\) above the horizontal. Find \(V\) and \(\theta\).

3
\includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-05_448_802_258_676} The diagram shows the cross-section through the centre of mass of a uniform solid object. The object is a cylinder of radius 0.2 m and length 0.7 m , from which a hemisphere of radius 0.2 m has been removed at one end. The point \(A\) is the centre of the plane face at the other end of the object. Find the distance of the centre of mass of the object from \(A\).
[0pt] [The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]

4 A small ball 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. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the ball.
2. Find \(x\) for the position of the ball when its path makes an angle of \(15 ^ { \circ }\) below the horizontal. [4]

5 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point \(( 0.5 + x ) \mathrm { m }\) vertically below \(O\). The particle \(P\) comes to instantaneous rest at \(O\).

1. Find \(x\).
2. Find the greatest speed of \(P\).

6
\(A B C\) is a uniform lamina in the form of a triangle with \(A B = 0.3 \mathrm {~m} , B C = 0.6 \mathrm {~m}\) and a right angle at \(B\) (see diagram).

1. State the distances of the centre of mass of the lamina from \(A B\) and from \(B C\). Distance from \(A B\)
   Distance from \(B C\) \(\_\_\_\_\)
   The lamina is freely suspended at \(B\) and hangs in equilibrium.
2. Find the angle between \(A B\) and the horizontal.
   A force of magnitude 12 N is applied along the edge \(A C\) of the lamina in the direction from \(A\) towards \(C\). The lamina, still suspended at \(B\), is now in equilibrium with \(A B\) vertical.
3. Calculate the weight of the lamina.

7 A particle \(P\) of mass 0.5 kg is attached to a fixed point \(O\) by a light elastic string of natural length 1 m and modulus of elasticity 16 N . The particle \(P\) is projected vertically upwards from \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 x ^ { 2 } \mathrm {~N}\) acts on \(P\) when \(P\) has displacement \(x \mathrm {~m}\) above \(O\). After projection the upwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, before the string becomes taut, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.2 x ^ { 2 }\).
2. Find the velocity of \(P\) at the instant the string becomes taut.
3. Find an expression for the acceleration of \(P\) while it is moving upwards after the string becomes taut.
4. Verify that \(P\) comes to instantaneous rest before the extension of the string is 0.5 m .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.