WME01 Mechanics 1 (M1): January 2025 Paper 01

A particle of mass 2.5 kg moves on a smooth horizontal plane under the action of three horizontal forces, F₁, F₂ and F₃, where

(a)Find the magnitude of the acceleration of the particle.[4]

A fourth force, F₄ = ( pi + pj) N, where p is a constant, is added.

The resultant of the four forces acts in the direction of the vector (7i + 2j).

(b)Find the value of p.[4]

The fixed points A, B and C lie in a straight line on a horizontal road.

• At time t = 0, a motorbike passes through A with speed 5 m s^–1

• From A, the motorbike accelerates uniformly until it reaches B with a speed of V m s^–1

• The motorbike takes T₁ seconds to travel from A to B

• From B, the motorbike decelerates uniformly until it comes to rest at C

• The motorbike takes T₂ seconds to travel from B to C

(a)Sketch a speed‑time graph for the motion of the motorbike as it moves from A to C.[3]

The distance AB is 132 m and the distance BC is 136 m.

(b)Find, in terms of V, an expression for

(i)T₁

(ii)T₂[4]

Given that the motorbike takes 28 s to travel from A to C,

(c)find the value of V,[2]

(d)find the deceleration of the motorbike.[2]

A particle P of mass 3m kg and a particle Q of mass m kg are at rest on a rough horizontal surface. The distance between P and Q is 1.4 m, as shown in Figure 1.

An impulse of magnitude λ N s is now applied to P in the direction PQ. Immediately after the impulse is applied, the speed of P is 5 m s^–1

(a)Find λ in terms of m.[2]

Immediately before P collides with Q, the speed of P is 2.5 m s^–1 The coefficient of friction between P and the surface is μ

(b)Find the value of μ[7]

Immediately after P collides with Q, the speed of Q is 2.1 m s^–1

(c)Find the speed of P immediately after the collision.[3]

A uniform rod AB has length 3 m and weight W newtons. The rod is suspended by two light vertical ropes. One rope is attached to the rod at A and the other rope is attached to the rod at C, where AC = 2.1 m.

The rod is in equilibrium in a horizontal position, as shown in Figure 2.

The tension in the rope at C is 350 N.

(a)Show that W = 490[3]

A particle P of weight 210 N is attached to the rod at a distance d metres from A. The tension in the rope at C is now 600 N. The rod remains in equilibrium in a horizontal position.

(b)Find the value of d.[3]

Particle P is removed from the rod. A particle Q of weight X newtons is now attached at B. The rod remains in equilibrium in a horizontal position and is now on the point of tilting.

(c)Find the value of X.[4]

[In this question i and j are horizontal perpendicular unit vectors and position vectors are given relative to a fixed origin.]

In a game, a ball B is rolled across a horizontal surface towards a fixed target. The ball is modelled as a particle moving with constant velocity.

At time t = 1 s, the position vector of B is (–2i + 5j) m. At time t = 9 s, the position vector of B is (10i – 3j) m.

(a)Find the velocity of the ball.[3]

The position vector of the target is (13i – 5j) m.

(b)Use the model to find the distance of B from the target at time t = 7 s.[4]

A particle P of mass m is held in equilibrium on a fixed rough inclined plane by a light inextensible string.

The plane is inclined at an angle α to the horizontal, where α < 45° The string is inclined to the plane at angle α, as shown in Figure 3.

The string lies in a vertical plane that contains a line of greatest slope of the inclined plane.

When the tension in the string is 0.75 mg, P is on the point of moving up the plane.

(a)Find an expression for the magnitude of the frictional force acting on P, giving your answer in terms of m, g and α.[3]

The coefficient of friction between P and the plane is 12

(b)Show that

[6]

The string breaks.

(c)Determine whether P remains at rest. You must justify your reasoning.[3]

One end of a light inextensible string is attached to a particle A of mass 2m. The other end is attached to a particle B of mass 3m. The string passes over a small smooth fixed pulley. The string is taut and both straight parts of the string are vertical. Both particles are held at a distance 0.5 m above a horizontal surface, as shown in Figure 4.

The system is released from rest and B moves downwards.

(a)Find the tension in the string in terms of m and g.[5]

(b)Find the speed of B at the instant it strikes the surface.[4]

In the subsequent motion, A does not reach the pulley and B does not rebound after it strikes the surface.

(c)Find the time from the instant when the system is released from rest to the instant when A first reaches a height of 1.06 m above the surface.[6]