CAIE M2 (Mechanics 2) 2014 November

A particle of mass \(m\) moves in a straight line. At time \(t\), its displacement from a fixed point on the line is \(s\) and its velocity is \(v\). The particle experiences a retarding force of magnitude \(mkv^2\), where \(k\) is a positive constant. Find the relationship between \(v\) and \(t\). [7]

\includegraphics{figure_2} A uniform rod \(AB\) of mass \(3m\) and length \(4a\) rests in equilibrium in a vertical plane with the end \(A\) on rough horizontal ground and the end \(B\) against a smooth vertical wall. The rod makes an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac{3}{5}\).

1. Find the normal reaction at \(A\) and the normal reaction at \(B\). [4]
2. Find the coefficient of friction between the rod and the ground. [2]

A particle \(P\) of mass \(0.2\) kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle moves in a horizontal circle of radius \(0.8\) m with the string making a constant angle of \(60°\) with the vertical. Calculate the speed of the particle and the tension in the string. [6]

\includegraphics{figure_4} The diagram shows the cross-section of a uniform solid consisting of a cylinder of radius \(0.4\) m and height \(1.5\) m with a hemisphere of radius \(0.4\) m on top.

1. Find the distance of the centre of mass above the base of the cylinder. [5]
2. The solid can just rest in equilibrium on a plane inclined at angle \(\alpha\) to the horizontal. Find \(\alpha\). [3]

The position vector of a particle at time \(t\) is given by \(\mathbf{r} = t^2\mathbf{i} + (3t - 1)\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. Find the velocity and acceleration of the particle when \(t = 2\).

1. Hence find the angle between the velocity and acceleration vectors when \(t = 2\). [3]
2. Find the value of \(t\) for which the velocity and acceleration vectors are perpendicular. [4]

A particle of mass \(2\) kg moves under the action of a variable force. At time \(t\) seconds the force is \((6t - 3)\mathbf{i} + 4\mathbf{j}\) newtons, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. When \(t = 0\), the particle is at rest at the origin.

1. Find the velocity of the particle when \(t = 4\). [4]
2. Find the kinetic energy of the particle when \(t = 4\). [2]
3. Find the distance of the particle from the origin when \(t = 2\). [6]

\includegraphics{figure_7} A particle of mass \(0.4\) kg is attached to one end of a light inextensible string of length \(2\) m. The other end of the string is attached to a fixed point \(O\). The particle moves in a vertical circle and passes through the lowest point of the circle with speed \(6\) m s\(^{-1}\).

1. Find the tension in the string when the particle is at the lowest point. [2]
2. Find the speed of the particle when the string makes an angle of \(60°\) with the downward vertical. [4]
3. Hence find the tension in the string at this position. [2]