WME01 Mechanics 1 (M1): January 2020 Paper 01

Two particles, P and Q, of mass m₁ and m₂ respectively, are moving on a smooth horizontal plane. The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, both particles are moving with speed u.

The direction of motion of each particle is reversed by the collision.

Immediately after the collision, the speed of Q is 13 u.

(a)Find, in terms of m₂ and u, the magnitude of the impulse exerted by P on Q in the collision.[3]

(b)Find, in terms of m₁, m₂ and u, the speed of P immediately after the collision.[3]

(c)Hence show that m₂ > 34 m₁[2]

A non‑uniform beam AB has length 6 m and weight W newtons. The beam is supported in equilibrium in a horizontal position by two vertical ropes, one attached to the beam at A and the other attached to the beam at C, where CB = 1.5 m, as shown in Figure 1.

The centre of mass of the beam is 2.625 m from A.

The ropes are modelled as light strings. The beam is modelled as a non‑uniform rod.

Given that the tension in the rope attached at C is 20 N greater than the tension in the rope attached at A,

(a)find the value of W.[6]

(b)State how you have used the fact that the beam is modelled as a rod.[1]

A particle, P, is projected vertically upwards with speed U from a fixed point O. At the instant when P reaches its greatest height H above O, a second particle, Q, is projected

with speed 12 U vertically upwards from O.

(a)Find H in terms of U and g.[2]

(b)Find, in terms of U and g, the time between the instant when Q is projected and the instant when the two particles collide.[6]

(c)Find where the two particles collide.[3]

Two identical small rings, A and B, each of mass m, are threaded onto a rough horizontal wire. The rings are connected by a light inextensible string. A particle C of mass 3m is attached to the midpoint of the string. The particle C hangs in equilibrium below the wire with angle BAC = β, as shown in Figure 2.

The tension in each of the parts, AC and BC, of the string is T

(a)By considering particle C, find T in terms of m, g and β[2]

(b)Find, in terms of m and g, the magnitude of the normal reaction between the wire and A.[3]

The coefficient of friction between each ring and the wire is 45

The two rings, A and B, are on the point of sliding along the wire towards each other.

(c)Find the value of tan β[5]

A car travels at a constant speed of 40 m s^–1 in a straight line along a horizontal racetrack. At time t = 0, the car passes a motorcyclist who is at rest. The motorcyclist immediately sets off to catch up with the car.

The motorcyclist accelerates at 4 m s^–2 for 15 s and then accelerates at 1 m s^–2 for a further T seconds until he catches up with the car.

(a)Sketch, on the same axes, the speed‑time graph for the motion of the car and the speed‑time graph for the motion of the motorcyclist, from time t = 0 to the instant when the motorcyclist catches up with the car.[2]

At the instant when t = t₁ seconds, the car and the motorcyclist are moving at the same speed.

(b)Find the value of t₁[2]

(c)Show that T²+ kT – 300 = 0, where k is a constant to be found.[6]

A force F is given by F = (10i + j) N.

(a)Find the exact value of the magnitude of F.[2]

(b)Find, in degrees, the size of the angle between the direction of F and the direction of the vector (i + j).[4]

The resultant of the force F and the force (–15i + aj) N, where a is a constant, is parallel to, but in the opposite direction to, the vector (2i – 3j).

(c)Find the value of a.[5]

A (4 kg) P

A particle A of mass 4 kg is held at rest on a rough horizontal table. Particle A is attached to one end of a string that passes over a pulley P. The pulley is fixed at the edge of the table. The other end of the string is attached to a particle B, of mass 3 kg, which hangs freely below P.

The part of the string from A to P is perpendicular to the edge of the table and A, P and B all lie in the same vertical plane.

The string is modelled as being light and inextensible and the pulley is modelled as being small, smooth and light.

The system is released from rest with the string taut. At the instant of release, A is 2 m from the edge of the table and B is 1.4 m above a horizontal floor, as shown in Figure 3.

After descending with constant acceleration for 2 seconds, B hits the floor and does not rebound.

(a)Show that the acceleration of A before B hits the floor is 0.7 m s^–2[2]

(b)State which of the modelling assumptions you have used in order to answer part (a).[1]

(c)Find the magnitude of the resultant force exerted on the pulley by the string.[4]

The coefficient of friction between A and the table is μ.

(d)Find the value of μ.[6]

(e)Determine, by calculation, whether or not A reaches the pulley.[5]