Edexcel A-Level Maths: Mechanics

A block $A$A of mass $3\ \text{kg}$3 kg rests on a rough plane inclined at $30^\circ$30∘ to the horizontal. The coefficient of friction between $A$A and the plane is $0.2$0.2. A light inextensible string runs from $A$A, up a line of greatest slope, over a smooth pulley fixed at the top of the plane, and down to a particle $B$B of mass $5\ \text{kg}$5 kg which hangs freely. The system is released from rest with the string taut, and $B$B descends while $A$A moves up the plane.

(a) Draw a clear force diagram for each particle and write down the equation of motion for $A$A along the plane and for $B$B. (4 marks)

(b) Find the acceleration of the system and the tension in the string. (5 marks)

(c) When $B$B has descended $1.5\ \text{m}$1.5 m the string suddenly breaks. Find how much further up the plane $A$A travels before it first comes instantaneously to rest. (3 marks)

(Take $g = 9.8\ \text{m s}^{-2}$g=9.8 m s−2.)

A stone is projected from a point $O$O at the top of a vertical cliff with speed $20\ \text{m s}^{-1}$20 m s−1 at an angle of $30^\circ$30∘ above the horizontal. The point $O$O is $25\ \text{m}$25 m vertically above horizontal ground. The stone is modelled as a particle moving freely under gravity.

(a) Find the time taken for the stone to reach its greatest height above $O$O. (3 marks)

(b) Find the total time the stone is in the air, and hence the horizontal distance of the point where it lands from the foot of the cliff. (4 marks)

(c) Find the speed of the stone and the direction in which it is moving at the instant it hits the ground. (3 marks)

(Take $g = 9.8\ \text{m s}^{-2}$g=9.8 m s−2.)

A particle $P$P moves along a straight line. At time $t$t seconds, where $t \geq 0$t≥0, the velocity of $P$P is

$v = 3t^2 - 12t + 9 \quad \text{m s}^{-1}.$v=3t2−12t+9m s−1.

When $t = 0$t=0 the particle is at the origin $O$O.

(a) Find the values of $t$t at which $P$P is instantaneously at rest, and find the acceleration of $P$P at each of these instants. (4 marks)

(b) Find the total distance travelled by $P$P in the first $3$3 seconds of its motion. (4 marks)

A train travels between two stations $P$P and $Q$Q on a straight horizontal track. Starting from rest at $P$P, it accelerates uniformly to a maximum speed of $30\ \text{m s}^{-1}$30 m s−1, reaching that speed $20\ \text{s}$20 s after leaving $P$P. It then travels at $30\ \text{m s}^{-1}$30 m s−1 for a time, before decelerating uniformly to rest at $Q$Q. The three phases of the motion are summarised below.

Phase	Initial speed	Final speed	Duration
Accelerating	$0\ \text{m s}^{-1}$0 m s−1	$30\ \text{m s}^{-1}$30 m s−1	$20\ \text{s}$20 s
Constant speed	$30\ \text{m s}^{-1}$30 m s−1	$30\ \text{m s}^{-1}$30 m s−1	$t_2$t2​
Decelerating	$30\ \text{m s}^{-1}$30 m s−1	$0\ \text{m s}^{-1}$0 m s−1	$t_3$t3​

The distance from $P$P to $Q$Q is $1800\ \text{m}$1800 m and the whole journey takes $90\ \text{s}$90 s.

Find the magnitude of the deceleration of the train in the final phase.

(6 marks)

A non-uniform rod $AB$AB has length $4\ \text{m}$4 m and mass $12\ \text{kg}$12 kg. It rests horizontally in equilibrium on two smooth supports at the points $C$C and $D$D, where $AC = 0.5\ \text{m}$AC=0.5 m and $AD = 3\ \text{m}$AD=3 m. The reaction of the support at $D$D on the rod is twice the reaction of the support at $C$C on the rod.

Find the distance of the centre of mass of the rod from $A$A.

(5 marks)

A box of mass $8\ \text{kg}$8 kg is pulled across rough horizontal ground by a constant horizontal force of magnitude $30\ \text{N}$30 N. The coefficient of friction between the box and the ground is $0.25$0.25. The box is modelled as a particle.

Find the acceleration of the box.

(4 marks)