OCR A Level Further Maths A Mechanics (optional - choose any two of Y542/Y543/Y544/Y545) June 2025 question paper Y543

A particle P of mass 4 kg is moving in a straight line with a speed of 7 m s^-1 on a smooth horizontal surface.

A constant force of magnitude 12 N acts on P for 5 seconds in the same direction as P ’s motion.

Find the magnitude of the impulse exerted by the force on P. You must include the units.

Hence determine the speed of P at the instant when the force stops acting.

By considering the change in the kinetic energy of P, determine the work done by the force

over the 5 seconds that it acts.

Hence find the distance travelled by P while the force is acting on it.

Dynamic viscosity is a physical property of liquids and is denoted by η.

The units of η are Ns /m².

Determine the dimensions of η.

Kinematic viscosity is another physical property of liquids and is denoted by ν. Kinematic viscosity and dynamic viscosity are related by the formula ν = where ρ is the density (mass per unit volume) of the liquid.

By finding the dimensions of ν, state a possible unit for ν.

In this question you must show detailed reasoning.

A particle P of mass 3.25 kg is moving on a smooth horizontal plane with velocity v ms^-1 where v = 4 i - 6 j.

Find the kinetic energy of P.

A system of horizontal forces starts to act on P. One of the forces is F N where F = i - (3 + t) j and t is the time, in seconds, after the system of forces starts to act.

Find the power developed by F at time t = 0.

Given that the velocity of P does not change at any time, find the total power developed by

the forces other than F at time t = 6.

Determine the value of e.

After this collision, Q goes on to collide directly with the wall. The coefficient of restitution between Q and the wall is .

Determine whether there will be a subsequent collision between P and Q.

One end of a light inextensible string of length 5 m is attached to a fixed point A. The other end is attached to a particle P of mass m kg which lies on a smooth horizontal plane.

With the string taut, P moves on the plane with constant speed v m s^-1 in a circular path. The centre of the circular path is the point O which is 4 m vertically below A. The string makes an angle of θ with the downward vertical through A (see diagram).

Given that the magnitude of the contact force between P and the surface is equal to the

tension in the string, determine the value of v.

Given instead that P is about to lose contact with the surface, determine the angular velocity

of P.

One end of a light inextensible string of length r m is attached to a fixed point O. The other end is attached to a particle P of mass m kg.

Particle P hangs in equilibrium vertically below O with the string taut. An initial horizontal impulse is applied to P so that it instantaneously starts moving horizontally with a speed of u m s^-1.

Particle P subsequently moves in complete circles in a vertical plane containing O.

When the line OP makes an angle of θ with the downward vertical through O, the speed of P is v m s^-1 and the tension in the string is TN.

Show that T = u² + α gr + β gr cos θ) where α and β are constants to be determined.

Determine, in terms of m, g and r, the minimum initial impulse required to ensure that P

describes complete vertical circles around O with the string remaining taut.

Show that while the force is acting the motion of P is modelled by the differential equation

Hence find v in terms of t, showing that this expression for v is valid for 0 ⩽ t ⩽ π.

Determine the total distance travelled by P in the interval 0 ⩽ t ⩽ 10. Give your answer in an

exact form.

Model 2 is a simplified model for the sign. In this model, the lamina is in the shape of a trapezium with vertices at the same points O, A, B and C as in Model 1 (see diagram).

The centre of mass of the lamina in Model 2 is at the point with coordinates ( .

Find the value of and the value of

The sign is placed on a rough plane inclined at 50° to the horizontal. The line OA rests on the plane with O below A. The line OA lies along a line of greatest slope of the plane.

You may assume that all forces acting on the sign lie in the vertical plane containing OA and the plane is sufficiently rough to prevent sliding.

You are given that and from Model 1 are both within 0.04 m of the actual coordinates of the centre of mass of the sign.

Determine whether the sign will topple. Justify your answer.

Use your answer to part (b) to explain whether Model 2 would have predicted the same

result as in part (c)(i).