CAIE M2 (Mechanics 2) 2015 June

1 A particle \(P\) of mass 0.6 kg is on the rough surface of a horizontal disc with centre \(O\). The distance \(O P\) is 0.4 m . The disc and \(P\) rotate with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis which passes through \(O\). Find the magnitude of the frictional force which the disc exerts on the particle, and state the direction of this force.

2 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 30 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) which hangs in equilibrium vertically below \(O\), with \(O P = 0.8 \mathrm {~m}\).

1. Show that the mass of \(P\) is 1.8 kg . The particle is pulled vertically downwards and released from rest from the point where \(O P = 1.2 \mathrm {~m}\).
2. Find the speed of \(P\) at the instant when the string first becomes slack.

3 A triangular frame \(A B C\) consists of two uniform rigid rods each of length 0.8 m and weight 3 N , and a longer uniform rod of weight 4 N . The triangular frame has \(A B = B C\), and angle \(B A C =\) angle \(B C A = 30 ^ { \circ }\).

1. Calculate the distance of the centre of mass of the frame from \(A C\). \includegraphics[max width=\textwidth, alt={}, center]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-2_722_335_1302_904} The vertex \(A\) of the frame is attached to a smooth hinge at a fixed point. The frame is held in equilibrium with \(A C\) vertical by a vertical force of magnitude \(F \mathrm {~N}\) applied to the frame at \(B\) (see diagram).
2. Calculate \(F\), and state the magnitude and direction of the force acting on the frame at the hinge.

4 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of weight 6 N . Another light inextensible string of length 0.5 m connects \(P\) to a fixed point \(B\) which is 0.8 m vertically below \(A\). The particle \(P\) moves with constant speed in a horizontal circle with centre at the mid-point of \(A B\). Both strings are taut.

1. Calculate the speed of \(P\) when the tension in the string \(B P\) is 2 N .
2. Show that the angular speed of \(P\) must exceed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-3_499_721_715_712} A uniform solid cube with edges of length 0.4 m rests in equilibrium on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \(A B C D\) is a cross-section through the centre of mass of the cube, with \(A B\) along a line of greatest slope. \(B\) lies below the level of \(A\). One end of a light elastic string with modulus of elasticity 12 N and natural length 0.4 m is attached to \(C\). The other end of the string is attached to a point below the level of \(B\) on the same line of greatest slope, such that the string makes an angle of \(30 ^ { \circ }\) with the plane (see diagram). The cube is on the point of toppling. Find

1. the tension in the string,
2. the weight of the cube.

6 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small ball \(B\) is projected with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal from a point \(O\). At time 2 s after the instant of projection, \(B\) strikes a smooth wall which slopes at \(60 ^ { \circ }\) to the horizontal. The speed of \(B\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its direction of motion is perpendicular to the wall at the instant of impact (see Fig. 1). \(B\) bounces off the wall with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the wall. At time 0.8 s after \(B\) bounces off the wall, \(B\) strikes the wall again at a lower point \(A\) (see Fig. 2).

1. Find \(U\) and \(\theta\).
2. By considering the motion of \(B\) after it bounces off the wall, calculate \(V\).

7 A force of magnitude \(0.4 t \mathrm {~N}\), applied at an angle of \(30 ^ { \circ }\) above the horizontal, acts on a particle \(P\), where \(t \mathrm {~s}\) is the time since the force starts to act. \(P\) is at rest on rough horizontal ground when \(t = 0\). The mass of \(P\) is 0.2 kg and the coefficient of friction between \(P\) and the ground is \(\mu\).

1. Given that \(P\) is about to slip when \(t = 2\), find \(\mu\) and the value of \(t\) for the instant when \(P\) loses contact with the ground.
2. While \(P\) is moving on the ground, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 2.165 t - 4.330$$ where the coefficients are correct to 4 significant figures.
3. Calculate the speed of \(P\) when it loses contact with the ground. {www.cie.org.uk} after the live examination series. }