WME01 Mechanics 1 (M1): Oct/Nov 2021 Paper 01

A non‑uniform rod AB has length 9 m and mass M kg.

The rod rests in equilibrium in a horizontal position on two supports, one at C where AC = 2.5 m and the other at D where DB = 2 m, as shown in Figure 1.

The magnitude of the force acting on the rod at D is twice the magnitude of the force acting on the rod at C.

The centre of mass of the rod is d metres from A.

Find the value of d.[6]

A particle P of mass 2m is moving on a rough horizontal plane when it collides directly with a particle Q of mass 4m which is at rest on the plane. The speed of P immediately before the collision is 3u. The speed of Q immediately after the collision is 2u.

(a)Find, in terms of u, the speed of P immediately after the collision.[3]

(b)State clearly the direction of motion of P immediately after the collision.[1]

Following the collision, Q comes to rest after travelling a distance 6ug along the plane.

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

(c)Find the value of μ.[6]

A car is moving at a constant speed of 25 m s^–1 along a straight horizontal road.

The car is modelled as a particle.

At time t = 0, the car is at the point A and the driver sees a road sign 48 m ahead.

Let t seconds be the time that elapses after the car passes A.

In a first model, the car is assumed to decelerate uniformly at 6 m s^–2 from A until the car reaches the road sign.

(a)Use this first model to find the speed of the car as it reaches the sign.[2]

The road sign indicates that the speed limit immediately after the sign is 13 m s^–1.

In a second model, the car is assumed to decelerate uniformly at 6 m s^–2 from A until it reaches a speed of 13 m s^–1. The car then maintains this speed until it reaches the road sign.

(b)Use this second model to find the value of t at which the car reaches the sign.[4]

In a third model, the car is assumed to move with constant speed 25 m s^–1 from A until time t = 0.2, the car then decelerates uniformly at 6 m s^–2 until it reaches a speed of 13 m s^–1. The car then maintains this speed until it reaches the road sign.

(c)Use this third model to find the value of t at which the car reaches the sign.[4]

The position vector, r metres, of a particle P at time t seconds, relative to a fixed origin O, is given by

(a)Find, to the nearest degree, the size of the angle between r and the vector j, when t = 2[3]

(b)Find the values of t for which the distance of P from O is 2.5 m.[5]

A small bead of mass 0.2 kg is attached to the end P of a light rod PQ. The bead is threaded onto a fixed vertical rough wire.

The bead is held in equilibrium with the rod PQ inclined to the wire at an angle α, where

tan α = 43 , as shown in Figure 2.

The thrust in the rod is T newtons.

The bead is modelled as a particle.

(a)Find the magnitude and direction of the friction force acting on the bead when T = 2.5[3]

The coefficient of friction between the bead and the wire is μ.

Given that the greatest possible value of T is 6.125

(b)find the value of μ.[7]

A small ball is thrown vertically upwards at time t = 0 from a point A which is above horizontal ground. The ball hits the ground 7 s later.

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

The velocity‑time graph shown in Figure 3 represents the motion of the ball for 0  t  7

(a)Find the speed with which the ball is thrown.[2]

(b)Find the height of A above the ground.[3]

One end of a light inextensible string is attached to a particle A of mass 2m. The other end of the string is attached to a particle B of mass 3m. The string passes over a small, smooth, light pulley P which is fixed at the top of a rough inclined plane. The plane is inclined to

the horizontal at an angle α, where tan α = 34

Particle A is held at rest on the plane with the string taut and B hanging freely below P, as shown in Figure 4. The section of the string AP is parallel to a line of greatest slope of the plane.

The coefficient of friction between A and the plane is 12

Particle A is released and begins to move up the plane.

For the motion before A reaches the pulley,

(a)(i)write down an equation of motion for A,

(ii)write down an equation of motion for B,[4]

(b)find, in terms of g, the acceleration of A,[5]

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

(d)State how you have used the information that P is a smooth pulley.[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.]

At 7 am a ship leaves a port and moves with constant velocity. The position vector of the port is (–2i + 9j) km.

At 7.36 am the ship is at the point with position vector (4i + 6j) km.

(a)Show that the velocity of the ship is (10i – 5j) km h^–1[2]

(b)Find the position vector of the ship t hours after leaving port.[2]

At 8.48 am a passenger on the ship notices that a lighthouse is due east of the ship.

At 9 am the same passenger notices that the lighthouse is now north east of the ship.

(c)Find the position vector of the lighthouse.[4]

(d)Find the position vector of the ship when it is due south of the lighthouse.[4]