CAIE M2 (Mechanics 2) 2016 November

1 A particle \(P\) of mass 0.3 kg moves in a circle with centre \(O\) on a smooth horizontal surface. \(P\) is attached to \(O\) by a light elastic string of modulus of elasticity 12 N and natural length \(l \mathrm {~m}\). The speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the radius of the circle in which it moves is \(2 l \mathrm {~m}\). Calculate \(l\).

2
\includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-2_318_495_484_824} A uniform wire is bent to form an object which has a semicircular arc with diameter \(A B\) of length 1.2 m , with a smaller semicircular arc with diameter \(B C\) of length 0.6 m . The end \(C\) of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.

1. Show that the distance of the centre of mass of the object from the line \(A C B\) is 0.191 m , correct to 3 significant figures. The object is freely suspended at \(A\) and hangs in equilibrium.
2. Find the angle between \(A C B\) and the vertical.

3 A small block \(B\) of mass 0.25 kg is released from rest at a point \(O\) on a smooth horizontal surface. After its release the velocity of \(B\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement is \(x \mathrm {~m}\) from \(O\). The force acting on \(B\) has magnitude \(\left( 2 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and is directed horizontally away from \(O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 8\).
2. Find the velocity of \(B\) when \(x = 1.5\). An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 6 - 3 x$$
3. Find the magnitude of this extra force and state the direction in which it acts.

4
\includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-3_388_650_264_749} The diagram shows the cross-section \(A B C D\) through the centre of mass of a uniform solid prism. \(A B = 0.9 \mathrm {~m} , B C = 2 a \mathrm {~m} , A D = a \mathrm {~m}\) and angle \(A B C =\) angle \(B A D = 90 ^ { \circ }\).

1. Calculate the distance of the centre of mass of the prism from \(A D\).
2. Express the distance of the centre of mass of the prism from \(A B\) in terms of \(a\). The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(A D\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(B C\).
3. Given that the prism is on the point of toppling, calculate \(a\).

5 A small ball \(B\) of mass 0.4 kg moves in a horizontal circle with centre \(O\) and radius 0.6 m on a smooth horizontal surface. One end of a light inextensible string is attached to \(B\); the other end of the string is attached to a fixed point 0.45 m vertically above \(O\).

1. Given that the tension in the string is 5 N , calculate the speed of \(B\).
2. Find the greatest possible tension in the string for the motion, and the corresponding angular speed of \(B\).

6
\includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-3_483_419_1800_863} The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre \(O\) and radius 0.9 m . A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point \(A\) on the inside of the tube. The other end of the string is attached to a particle \(P\) of mass 0.2 kg . The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate

1. the speed of \(P\) when it is at the same horizontal level as \(O\),
2. the speed of \(P\) at the instant when the string becomes slack.

7 A particle \(P\) is projected with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a horizontal plane. In the subsequent motion, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is $$y = k x - \frac { \left( 1 + k ^ { 2 } \right) x ^ { 2 } } { 245 }$$ where \(k\) is a constant. \(P\) passes through the points \(A ( 14 , a )\) and \(B ( 42,2 a )\), where \(a\) is a constant.

1. Calculate the two possible values of \(k\) and hence show that the larger of the two possible angles of projection is \(63.435 ^ { \circ }\), correct to 3 decimal places. For the larger angle of projection, calculate
2. the time after projection when \(P\) passes through \(A\),
3. the speed and direction of motion of \(P\) when it passes through \(B\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }