Edexcel A Level Further Maths M5 · Mechanics 5 June 2012 question paper 6681

A particle P moves in a plane such that its position vector r metres at time ¢ seconds

Find r in terms of (9)

upwards from rest. The rocket burns fuel at a rate of 15 kg s! and the burnt fuel is ejected vertically downwards with a speed of 1000 m relative to the rocket. At time ¢ seconds

Show that

Auniform rod PQ, ofmass m and length 3a, is free to rotate about a fixed smooth horizontal axis L, which passes through the end P of the rod and is perpendicular to the rod. The rod hangs at rest in equilibrium with Q vertically below P. One end of a light inextensible string of length 2a is attached to the rod at P and the other end is attached to a particle of mass 3m. The particle is held with the string taut, and horizontal and perpendicular to L, and is then released. After colliding, the particle sticks to the rod forming a body B.

Show that the moment of inertia of B about L is 15ma?. (2)

Show that B first comes to instantaneous rest after it has turned through an angle

A body consists of a uniform plane circular disc, of radius r and mass 2m, with a particle of mass 3m attached to the circumference of the disc at the point P. The line PQ is a diameter of the disc. The body is free to rotate in a vertical plane about a fixed smooth horizontal axis, L, which is perpendicular to the plane of the disc and passes through Q. The body is held with OP making an angle £ with the downward

component, perpendicular to PQ, of the force acting on the body at Q at the instant when it is released. [You may assume that the momentof inertia of the body about L is (6)

relative to a fixed origin O.

7 N, 3 N and 3V10 N respectively.

Express F,, F, and F, in vector form. (3)

Show that the resultant of F, and F, is (2i— 10j — 16k) N. (2)

Find a vector equation of the line of action of this resultant, giving your in

circular pulley, of mass 4m and radius r, is free to rotate about a fixed smooth horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. A light inextensible string passes over the pulley and hasa particle of mass 2m attached to one end anda particle of mass 3m attached to the other end. The particles hang with the string vertical and taut on each side of the pulley. The rim of the pulley is sufficiently rough to prevent the string slipping. The system is released from rest.

Find the angular acceleration of the pulley. (8) When the angular speed of the pulley is Q, the string breaks and a constant braking couple of magnitude G is applied to the pulley which brings it to rest.

Find an expression for the angle turned through by the pulley from the instant when the string breaks to the instant when the pulley first comes to rest. (4)

A uniform lamina of mass m is in the shape of a triangle ABC. The perpendicular distance of C from the line AB is h. Prove, using integration, that the moment of inertia of the lamina about AB is mh?. (7)

Deduce the radius of gyration of a uniform square lamina of side 2a, about a diagonal.(3) The points X and Y are the mid-points of the sides RO and RS respectively of a square PORS of side 2a. A uniform lamina of mass M is in the shape of POXYS.

Show that the moment of inertia of this lamina about XY is Ma’. (6)