WME01 Mechanics 1 (M1): Oct/Nov 2025 Paper 01

Figure 1 shows a sketch of a beam AB, with weight 240 N and length 8 m.

The beam is held in equilibrium in a horizontal position by two vertical ropes. The ropes are attached to the beam at the points C and D, where AC = 2 m and DB = x metres.

The beam is modelled as a uniform rod and the ropes are modelled as light inextensible strings.

The tension in the rope at D is 90 N.

(a)Show that x = 23[3]

The rope at C will break if its tension exceeds 183 N. The rope at D cannot break. A package of weight W newtons is now attached to the beam at A. The beam remains horizontal and in equilibrium.

The package is modelled as a particle.

It is given that the rope at C does not break.

(b)Find the greatest possible value of W.[4]

Particle P of mass 7m and particle Q of mass 3m are moving in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly.

Immediately before the collision, the speed of P is ku and the speed of Q is u, as shown in Figure 2.

Immediately after the collision, the speed of P is w and the speed of Q is 2w.

The direction of motion of Q is reversed by the collision.

The impulse received by Q in the collision has magnitude 72 mu.

(a)Find w in terms of u.[3]

(b)Find the two possible values of k.[5]

A particle P of mass 2 kg moves on a smooth horizontal surface under the action of two forces F₁ and F₂, where F₁= (–2i + 3j) N and F₂= (4i + 2j) N.

(a)Find the acceleration of P.[3]

At time t = 0, the velocity of P is (3i – 4j) m s^–1

(b)Find the speed of P when t = 3 seconds.[4]

An additional force, F₃= (bi + cj) N, is applied to P.

The resultant of F₁, F₂ and F₃ is equal to λ(i + j) N, where λ is a constant.

(c)Show that b – c = 3[3]

The resultant of F₁, F₂ and F₃ has magnitude 10 2 N.

(d)Find the two possible F₃ forces.[4]

The point A is 10 m above horizontal ground. At time t = 0, a particle P is projected vertically upwards with speed 5 m s^–1 from A. Particle P moves freely under gravity.

(a)Find the greatest height above A reached by P.[3]

The point B is on the ground, vertically below A. At time t = 1 second, a particle Q is projected vertically upwards with speed 7 m s^–1 from B. Particle Q moves freely under gravity.

Particles P and Q collide at time t = T seconds.

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

(c)Find the speed of P at the instant immediately before the particles collide.[2]

A box of mass 4 kg is placed on a rough horizontal surface.

A force of magnitude P newtons, acting at 30° to the horizontal, is applied to the box, as shown in Figure 3.

The coefficient of friction between the box and the surface is 23

The box is modelled as a particle.

(a)Find the value of P when the box is on the point of sliding along the surface.[6]

The value of P is now increased to 25 and the box moves along the surface.

Find

(b)the acceleration of the box,[5]

(c)the speed of the box when it has moved 1.5 m.[2]

The point O lies on a straight horizontal road. At time t = 0, a car leaves O and travels along the road.

The velocity-time graph in Figure 4 shows the velocity, v m s^–1, of the car at time t seconds for the first 35 seconds of its journey.

(a)Find

(i)the acceleration of the car for the period 0  t  10

(ii)the deceleration of the car for the period 30  t  35[2]

(b)Sketch an acceleration-time graph for the car for the period 0  t  35[2]

(c)Find the distance travelled by the car for the period 0  t  35[2]

When t = 5, a motorcycle starts from rest at O. The motorcycle travels along the same road as the car and in the same direction.

For the period 5  t  20, the acceleration of the motorcycle is A m s^–2, where A is a positive constant.

The motorcycle catches up with the car when t = 20

(d)Find the value of A.[4]

A block P of mass 2m is held at rest on a fixed rough plane.

The plane is inclined to the horizontal at an angle α, where tan α = 512

One end of a light inextensible string is attached to P. The string is parallel to a line of greatest slope of the plane and passes over a smooth light pulley which is fixed at the top of the plane. The other end of the string is attached to a block Q of mass 5m. Block Q hangs vertically below the pulley, as shown in Figure 5.

The system is released from rest with the string taut and block P moves up the plane.

Immediately after the system is released, the tension in the string is T and the acceleration of the blocks is a.

The blocks are modelled as particles and air resistance is ignored.

(a)Write down an equation of motion for Q.[2]

The coefficient of friction between P and the plane is 18

(b)Find T in terms of m and g.[7]

(c)State how the solution to part (b) uses the fact that the string is inextensible.[1]

The magnitude of the force exerted on the pulley by the string is kmg.

(d)Find the value of k to 3 significant figures.[4]