1
A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $2a$. The other end of the string is attached to a fixed point $O$. The particle moves with constant speed $v$ in a horizontal circle, with the string inclined at an angle of $60°$ to the vertical. The centre of the circle is vertically below $O$. [3]
Find $v$ in terms of $a$ and $g$. [3]
Total for Question 1: 3 marks
2
A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $2mg$. The other end of the string is attached to a fixed point $O$ on a rough plane inclined at an angle of $30°$ to the horizontal. The particle is released from rest at $O$ and moves down the plane. The coefficient of friction between $P$ and the plane is $\dfrac{1}{3}$.
(a) Find the distance that $P$ moves down the plane before first coming to instantaneous rest. [5]
(b) State, with a reason, whether $P$ remains at rest in this position. [1]
Total for Question 2: 6 marks
3
The uniform triangular lamina $ABC$ is equilateral with side length $4a$. The lamina has mass $m$. A particle of mass $m$ is attached to the lamina at the midpoint of $BC$.
(a) Find the distance of the centre of mass of the combined object from $AB$. [3]
(b) The object is freely suspended from $A$ and hangs in equilibrium. Find the angle which $AC$ makes with the downward vertical. [4]
Total for Question 3: 7 marks
4
A hollow sphere of radius $a$ is fixed with its centre at the point $O$. A particle $P$ of mass $m$ moves on the smooth inner surface of the sphere in a horizontal circle with constant speed $\sqrt{\dfrac{3ag}{2}}$. The plane of the circle is at a distance $a\cos\theta$ below $O$, where $\theta$ is the angle between $OP$ and the upward vertical through $O$.
(a) Show that $\theta = 60°$. [3]
(b) Find the magnitude of the normal reaction between $P$ and the sphere in terms of $m$ and $g$. [2]
(c) Find the radius of the horizontal circle in terms of $a$. [2]
Total for Question 4: 7 marks
5
A particle $P$ of mass $1\;\text{kg}$ is projected vertically upwards from horizontal ground with speed $15\;{\mathrm{m\,s}}^{-1}$. The particle moves under gravity and is subject to a resistive force of magnitude $0.5v\;\text{N}$, where $v\;{\mathrm{m\,s}}^{-1}$ is the speed of $P$ at time $t$ seconds after projection.
(a) Show that $P$ comes to instantaneous rest after time $2\ln\dfrac{7}{4}\;\text{s}$. [4]
(b) Find the greatest height reached by $P$ above the point of projection. [4]
Total for Question 5: 8 marks
6
Two smooth spheres $A$ and $B$ lie on a smooth horizontal surface. Sphere $A$ has mass $3m$ and sphere $B$ has mass $m$. Sphere $A$ is moving with speed $2u$ in a direction making an angle $\alpha$ with the line of centres $AB$. Sphere $B$ is at rest. The spheres collide. The coefficient of restitution between the spheres is $\dfrac{1}{3}$. After the collision, $A$ moves in a direction making an angle $\varphi$ with $AB$.
(a) Show that $\tan\varphi = \dfrac{3}{2}\tan\alpha$. [4]
(b) Find the speed of $B$ after the collision in terms of $u$ and $\alpha$. [4]
Total for Question 6: 8 marks
7
A particle $P$ is projected from a point $O$ on horizontal ground with speed $20\;{\mathrm{m\,s}}^{-1}$ at an angle of $60°$ above the horizontal. The particle moves freely under gravity until it strikes the ground at point $A$. It immediately rebounds from the ground. The coefficient of restitution between $P$ and the ground is $\dfrac{1}{2}$.
(a) Find the distance $OA$. [2]
(b) Show that the distance $AB$ satisfies $AB = e \cdot OA$. [2]
(c) Find the speed of $P$ immediately after it rebounds at $A$. [3]
(d) Find the greatest height reached by $P$ after it rebounds at $A$, and find the total time from the instant of projection until $P$ reaches this greatest height. [4]
Total for Question 7: 11 marks