OCR A Level Further Maths A Mechanics (optional - choose any two of Y542/Y543/Y544/Y545) June 2023 question paper Y543

Show that at this instant the speed of P is 4.5 ms^-1.

Find the magnitude and direction of the radial acceleration of P at this instant.

Find the magnitude of the tangential acceleration of P at this instant.

Materials have a measurable property known as the Young’s Modulus, E.

If a force is applied to one face of a block of the material then the material is stretched by a

of the block.

Show that Strain is a dimensionless quantity.

By considering the dimensions of both Stress and Strain determine the dimensions of E.

It is suggested that the speed of sound in a material, c, depends only upon the value of Young’s modulus for the material, E, the volume of the material, V, and the density (or mass per unit volume) of the material, ρ.

Use dimensional analysis to suggest a formula for c in terms of E, V and ρ.

The speed of sound in a certain material is 500 ms^-1.

Use your formula from part (c) to predict the speed of sound in the material if the value

of Young’s modulus is doubled but all other conditions are unchanged.

With reference to your formula from part (c), comment on the effect on the speed of

sound in the material if the volume is doubled but all other conditions are unchanged.

Suggest one possible limitation caused by using dimensional analysis to set up the model in

part (c).

Two smooth circular discs A and B are moving on a smooth horizontal plane when they collide. The mass of A is 5 kg and the mass of B is 3 kg.

At the instant before they collide,

• the velocity of A is 4 ms^-1 at an angle of 60° to the line of centres,

• the velocity of B is 6 ms^-1 along the line of centres

(see diagram).

The coefficient of restitution for collisions between the two discs is

Determine the angle that the velocity of A makes with the line of centres after the collision.

ABCD is a uniform lamina in the shape of a kite with BA = BC = 0.37 m, DA = DC = 0.91 m and AC = 0.7 m (see diagram). The centre of mass of ABCD is G.

Explain why G lies on BD.

Show that the distance of G from B is 0.36 m.

The lamina ABCD is freely suspended from the point A.

Determine the acute angle that CD makes with the horizontal, stating which of C or D is

higher.

A particle P of mass 2 kg moves along the x-axis.

At time t = 0, P passes through the origin O with speed 3 ms^-1.

At time t seconds the displacement of P from O is x m and the velocity of P is v ms^-1, where t ⩾ 0, x ⩾ 0 and v ⩾ 0.

While P is in motion the only force acting on P is a resistive force F of magnitude (v² + 1) N acting in the negative x-direction.

Find an expression for v in terms of x.

Determine the distance travelled by P while its speed drops from 3 ms^-1 to 2 ms^-1.

Particle Q is identical to particle P. At a different time, Q is moving along the x-axis under the influence of a single constant resistive force of magnitude 1 N. When t ′ = 0, Q is at the origin and its speed is 3 ms^-1.

By comparing the motion of P with the motion of Q explain why P must come to rest at some

finite time when t < 6 with x < 9.

Sketch the velocity-time graph for P. You do not need to indicate any values on your sketch.

Determine the maximum displacement of P from O during P ’s motion.

A particle P of mass 3 kg is moving on a smooth horizontal surface under the influence of a variable horizontal force F N. At time t seconds, where t ⩾ 0, the velocity of P, v ms^-1, is given by

v = (32 sinh (2 t)) i + (32 cosh (2 t) - 257) j.

By considering kinetic energy, determine the work done by F over the interval

0 ⩽ t ⩽ ln 2.

Explain the significance of the sign of the answer to part (a)(i).

Determine the rate at which F is working at the instant when P is moving parallel to the

i-direction.

Determine the angular speed of A and the angular speed of B.

At the start of the motion, A, O and B all lie in the same vertical plane.

Find the first subsequent time when A, O and B all lie in the same vertical plane.

One end of a light elastic string of natural length 2.1 m and modulus of elasticity 4.8 N is attached to a particle, P, of mass 1.75 kg. The other end of the string is attached to a fixed point, O, which is on a rough inclined plane. The angle between the plane and the horizontal is θ where sin θ = The coefficient of friction between P and the plane is 0.732.

Particle P is placed on the plane at O and then projected down a line of greatest slope of the plane with an initial speed of 2.4 ms^-1.

Determine the distance that P has travelled from O at the instant when it first comes to rest. You

can assume that during its motion P does not reach the bottom of the inclined plane.