OCR GCE AS Level Further Maths B (MEI) Mechanics a (optional - choose any two of Y411/Y412/Y413/Y414) June 2022 question paper Y411

Fig. 1.1 and Fig. 1.2 show rigid rods with forces acting as marked. The diagrams are to scale, and in each figure the side length of a grid square is 1 metre.

• On the copy of Fig. 1.1 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents an anti‑clockwise couple of magnitude 24 N m.

• On the copy of Fig. 1.2 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents a clockwise couple of magnitude 1 N m.

Fig. 1.3 shows a rectangular lamina with two coplanar forces acting as marked. Each grid square has side length 1 m.

A third coplanar force, of magnitude T N, acts at A so that the resultant force on the lamina is zero.

Calculate the value of T.

Determine the magnitude and direction of the couple represented by this system of three

forces.

Three forces, of magnitudes 33 N, 45 N and P N, act at a point in the directions shown in the diagram. The system is in equilibrium.

Draw a triangle of forces for the system shown above. Your diagram should include the

magnitudes of the forces (33 N, 45 N and P N) and angle θ.

If P = 38, find, in degrees, the value of θ.

If θ = 40 °, determine the possible values for P.

Show that the triangular lamina has a mass of 3.3 kg.

Determine the value of x, correct to 3 significant figures.

One end of a light inextensible string is attached to S at D. The other end is attached to a fixed point on a vertical wall. For S to hang in equilibrium with AD vertical, a force of magnitude Q N is applied to S as shown in Fig. 3.2. The line of action of this force lies in the same plane as S. The string is taut and lies in the same plane as S at an angle φ to the downward vertical.

By taking moments about D, show that Q = 50.5, correct to 3 significant figures.

Determine, in degrees, the value of φ.

The diagram shows two points A and B on a snowy slope. A is a vertical distance of 25 m above B.

A rider and snowmobile, with a combined mass of 240 kg, start at the top of the slope, heading in the direction of B. As the snowmobile passes A, with a speed of 3 m s^-1, the rider switches off the engine so that the snowmobile coasts freely. When the snowmobile passes B, it has a speed of 18 m s^-1.

The resistances to motion can be modelled as a single, constant force of magnitude 120 N.

Calculate the distance the snowmobile travels from A to B.

Determine the maximum speed with which the rider and snowmobile can ascend.

The power developed by a vehicle is sometimes given in the non‑SI unit mechanical horsepower (hp). 1 hp is the power required to lift 550 pounds against gravity, starting and ending at rest, by 1 foot in 1 second.

Given that 1 metre ≈ 3.28 feet and 1 kg ≈ 2.2 pounds, determine the number of watts that are

equivalent to 1 hp.

Fig. 5.1 shows a small smooth sphere A at rest on a smooth horizontal surface. At both ends of the surface is a smooth vertical wall.

Sphere A is projected directly towards the left‑hand wall at a speed of 5 m s^-1. Sphere A collides directly with the left‑hand wall, rebounds, then collides directly with the right‑hand wall. After this second collision A has a speed of 3.2 m s^-1.

Explain how it can be deduced that the collision between A and the left‑hand wall was not

inelastic.

The coefficient of restitution between A and each wall is e.

Calculate the value of e.

When B is in limiting equilibrium, show that tan α =

If α = 40 °, determine the acceleration of B down the slope.

A horizontal force of magnitude P N is now applied to B, as shown in the diagram below. At first B is at rest.

P is gradually increased.

Show that, for B to slide on the slope,

Determine, in degrees, the least value of α for which B will not slide no matter how large P

becomes.

The diagram shows a cannon fixed to a trolley. The trolley runs on a smooth horizontal track.

A driver boards the trolley with two cannon balls. The combined mass of the trolley, driver,

cannon and cannon balls is 320 kg. Each cannon ball has a mass of 5 kg. Initially the trolley is at rest.

A force of 480 N acts on the trolley in the forward direction for 4 seconds.

Calculate the magnitude of the impulse of the force on the trolley.

Calculate the speed of the trolley after the force stops acting.

The driver now fires a cannon ball horizontally in the backward direction. The cannon ball and cannon separate at a rate of 90 m s^-1.

Show that, after the firing of the cannon ball, the trolley moves with a speed of 7.41 m s^-1

correct to 3 significant figures.

The driver now reverses the direction of the cannon and fires the second cannon ball horizontally in the forward direction. Again, the cannon ball and cannon separate at a rate of 90 m s^-1.

Calculate the overall percentage change in the kinetic energy of the trolley (alone) from before the first cannon ball is fired to after the second is fired, giving your answer correct to 2 decimal places. You should make clear whether the change in kinetic energy is a gain or a

loss.

Give a reason why one of the modelling assumptions that was required in answering parts

(a), (b) and (c) may not have been appropriate.