CAIE M2 (Mechanics 2) 2016 June

1 A small ball \(B\) is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 0.8 s after projection, \(B\) is 0.5 m vertically above the top of a vertical post.

1. Calculate the height of the top of the post above the ground.
2. Show that \(B\) is at its greatest height 0.2 s before passing over the post.

2 One end of a light elastic string of natural length 0.4 m is attached to a fixed point \(O\). The other end of the string is attached to a particle of weight 5 N which hangs in equilibrium 0.6 m vertically below \(O\).

1. Find the modulus of elasticity of the string. The particle is projected vertically upwards from the equilibrium position and comes to instantaneous rest after travelling 0.3 m upwards.
2. Calculate the speed of projection of the particle.
3. Calculate the greatest extension of the string in the subsequent motion.

3 The point \(O\) is 8 m above a horizontal plane. A particle \(P\) is projected from \(O\). After projection, 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 $$y = 2 x - x ^ { 2 }$$

1. Find the value of \(x\) for the point where \(P\) strikes the plane.
2. Find the angle and speed of projection of \(P\).
3. Calculate the speed of \(P\) immediately before it strikes the plane.

4
\includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-3_784_556_260_790} A uniform object is made by drilling a cylindrical hole through a rectangular block. The axis of the cylindrical hole is perpendicular to the cross-section \(A B C D\) through the centre of mass of the object. \(A B = C D = 0.7 \mathrm {~m} , B C = A D = 0.4 \mathrm {~m}\), and the centre of the hole is 0.1 m from \(A B\) and 0.2 m from \(A D\) (see diagram). The hole has a cross-section of area \(0.03 \mathrm {~m} ^ { 2 }\).

1. Show that the distance of the centre of mass of the object from \(A B\) is 0.212 m , and calculate the distance of the centre of mass from \(A D\). The object has weight 70 N and is placed on a rough horizontal surface, with \(A D\) in contact with the surface. A vertically upwards force of magnitude \(F \mathrm {~N}\) acts on the object at \(C\). The object is on the point of toppling.
2. Find the value of \(F\). The force acting at \(C\) is removed, and the object is placed on a rough plane inclined at an angle \(\theta ^ { \circ }\) to the horizontal. \(A D\) lies along a line of greatest slope, with \(A\) higher than \(D\). The plane is sufficiently rough to prevent sliding, and the object does not topple.
3. Find the greatest possible value of \(\theta\).

5 A particle \(P\) of mass 0.4 kg is placed at rest at a point \(A\) on a rough horizontal surface. A horizontal force, directed away from \(A\) and with magnitude \(0.6 t \mathrm {~N}\), acts on \(P\), where \(t \mathrm {~s}\) is the time after \(P\) is placed at \(A\). The coefficient of friction between \(P\) and the surface is 0.3 , and \(P\) has displacement from \(A\) of \(x \mathrm {~m}\) at time \(t \mathrm {~s}\).

1. Show that \(P\) starts to move when \(t = 2\). Show also that when \(P\) is in motion it has acceleration \(( 1.5 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Express the velocity of \(P\) in terms of \(t\), for \(t \geqslant 2\).
3. Express \(x\) in terms of \(t\), for \(t \geqslant 2\).

6
\includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-4_479_499_255_824}
\(O A\) is a rod which rotates in a horizontal circle about a vertical axis through \(O\). A particle \(P\) of mass 0.2 kg is attached to the mid-point of a light inextensible string. One end of the string is attached to the \(\operatorname { rod }\) at \(A\) and the other end of the string is attached to a point \(B\) on the axis. It is given that \(O A = O B\), angle \(O A P =\) angle \(O B P = 30 ^ { \circ }\), and \(P\) is 0.4 m from the axis. The rod and the particle rotate together about the axis with \(P\) in the plane \(O A B\) (see diagram).

1. Calculate the tensions in the two parts of the string when the speed of \(P\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angular speed of the rod is increased to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and it is given that the system now rotates with angle \(O A P =\) angle \(O B P = 60 ^ { \circ }\).
2. Show that the tension in the part \(A P\) of the string is zero. \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. }