CAIE M2 (Mechanics 2) 2013 November

A particle moves in a straight line. Its displacement from a fixed point \(O\) at time \(t\) seconds is \(s\) metres, where \(s = t^3 - 9t^2 + 24t\).

1. Find expressions for the velocity \(v\) and acceleration \(a\) of the particle at time \(t\).
2. Find the values of \(t\) for which the particle is at rest.
3. Find the total distance travelled by the particle in the first \(6\) seconds.

[8]

A particle moves in a straight line. At time \(t\) seconds its velocity is \(v\) ms\(^{-1}\) and its acceleration is \(a\) ms\(^{-2}\).

1. Given that \(a = —\), express \(v\) in terms of \(t\).
2. Given that \(v = tv\) when \(t = 0\), find \(v\) in terms of \(t\).
3. Find the displacement from the starting point when \(t = v\).

[6]

\includegraphics{figure_3} A particle moves on the inner surface of a smooth hollow cone of semi-vertical angle \(\alpha\). The axis of the cone is vertical with the vertex at the bottom. The particle moves in a horizontal circle of radius \(r\) with constant speed \(v\). Find expressions for the normal reactions on the particle from the cone surface, and show that the height of the particle above the vertex is \(\frac{v^2}{g \tan \alpha}\). [8]

A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point. The particle moves in a vertical circle.

1. Show that the speed \(v\) at the lowest point of the circle must satisfy \(v^2 \geq 5gl\) for the particle to complete the circle.
2. Given that the particle just completes the circle, find the tensions in the string at the highest and lowest points of the circle.
3. Given that \(v^2 = 6gl\) at the lowest point, find the tension in the string when the particle has risen through an angle \(\theta\) from the lowest point.

[14]

A smooth sphere of mass \(M\) and radius \(a\) rests in contact with a smooth vertical wall and a smooth inclined plane. The plane makes an angle \(\alpha\) with the horizontal.

1. Find the magnitude of each of the contact forces acting on the sphere.
2. Find the range of values of \(\alpha\) for which this equilibrium is possible.

[8]

Two particles \(A\) and \(B\) have masses \(3m\) and \(2m\) respectively. Initially \(A\) is at rest and \(B\) is moving with speed \(u\) in a straight line towards \(A\). The coefficient of restitution between the particles is \(e\).

1. Find the speeds of the particles immediately after the collision.
2. Find the condition on \(e\) for \(A\) to be moving faster than \(B\) after the collision.

[8]

\includegraphics{figure_7} A uniform solid hemisphere of mass \(M\) and radius \(a\) is placed with its curved surface on rough horizontal ground. A horizontal force \(P\) is applied to the hemisphere at the centre of its flat circular face.

1. Find the minimum value of the coefficient of friction \(\mu\) between the hemisphere and the ground for the hemisphere to slide without toppling.
2. Show that if \(\mu < \frac{3}{8}\), the hemisphere will topple.
3. Find the maximum horizontal distance that the centre of mass of the hemisphere moves before toppling begins, given that \(\mu = \frac{1}{4}\) and the hemisphere starts from rest.
4. Find the angular acceleration of the hemisphere about its point of contact with the ground at the instant when toppling begins.

[16]