WME01 Mechanics 1 (M1): January 2022 Paper 01

A particle P of weight 5 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point O. The particle P is held in equilibrium by a force of magnitude F newtons. The direction of this force is perpendicular to the string and OP makes an angle of 60° with the vertical, as shown in Figure 1.

Find

(a)the value of F[3]

(b)the tension in the string.[3]

A particle P has mass km and a particle Q has mass m. The particles are moving towards each other in opposite directions along the same straight line when they collide directly.

Immediately before the collision, P has speed 3u and Q has speed u.

As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.

(a)Find the value of k.[4]

(b)Find, in terms of m and u, the magnitude of the impulse exerted on Q in the collision.[3]

A beam ADCB has length 5 m. The beam lies on a horizontal step with the end A on the step and the end B projecting over the edge of the step. The edge of the step is at the point D where DB = 1.3 m, as shown in Figure 2.

When a small boy of mass 30 kg stands on the beam at C, where CB = 0.5 m, the beam is on the point of tilting.

The boy is modelled as a particle and the beam is modelled as a uniform rod.

(a)Find the mass of the beam.[3]

A block of mass X kg is now placed on the beam at A.

The block is modelled as a particle.

(b)Find the smallest value of X that will enable the boy to stand on the beam at B without the beam tilting.[3]

(c)State how you have used the modelling assumption that the block is a particle in your calculations.[1]

At time t = 0, a small ball is projected vertically upwards from a point A which is 24.5 m above the ground. The ball first comes to instantaneous rest at the point B, where AB = 19.6 m and first hits the ground at time t = T seconds.

The ball is modelled as a particle moving freely under gravity.

(a)Find the value of T.[6]

(b)Sketch a speed-time graph for the motion of the ball from t = 0 to t = T seconds.

(No further calculations are needed in order to draw this sketch.)[2]

A particle of mass m rests in equilibrium on a fixed rough plane under the action of a force of magnitude X. The force acts up a line of greatest slope of the plane, as shown in Figure 3.

The plane is inclined at an angle α to the horizontal, where tan α = 34

The coefficient of friction between the particle and the plane is μ.

• When X = 2P, the particle is on the point of sliding up the plane.

• When X = P, the particle is on the point of sliding down the plane.

Find the value of μ.[8]

[In this question i and j are horizontal unit vectors.]

A particle P of mass 2 kg moves under the action of two forces, ( pi + qj) N and (2qi + pj) N , where p and q are constants.

Given that the acceleration of P is (i − j) m s⁻²

(a)find the value of p and the value of q.[5]

(b)Find the size of the angle between the direction of the acceleration and the vector j.[2]

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

At t = T seconds, P is moving in the direction of the vector (11i − 13j).

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

A particle P of mass 4m lies on the surface of a fixed rough inclined plane.

The plane is inclined to the horizontal at an angle α where tan α = 34

The particle P is attached to one end of a light inextensible string.

The string passes over a small smooth pulley that is fixed at the top of the plane. The other end of the string is attached to a particle Q of mass m which lies on a smooth horizontal plane.

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

The system is released from rest with the string taut, as shown in Figure 4, and P moves down the plane.

The coefficient of friction between P and the plane is 14

For the motion before Q reaches the pulley

(a)write down an equation of motion for Q,[1]

(b)find, in terms of m and g, the tension in the string,[7]

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

(d)State where in your working you have used the information that the string is light.[1]

[In this question i and j are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin.]

A ship A moves with constant velocity (3i – 10j) km h⁻¹

At time t hours, the position vector of A is r km.

At time t = 0, A is at the point with position vector (13i + 5j) km.

(a)Find r in terms of t.[2]

Another ship B moves with constant velocity (15i + 14j) km h⁻¹

At time t = 0, B is at the point with position vector (3i – 5j) km.

(b)Show that, at time t hours,

[4]

Given that the two ships do not change course,

(c)find the shortest distance between the two ships,[6]

(d)find the bearing of ship B from ship A when the ships are closest.[2]