Further Maths B (MEI) · OCR H635 · Mechanics a (optional - choose any two of Y411/Y412/Y413/Y414) · June 2024
Two horizontal forces of magnitudes 7 N and 15 N act at a point O. The 15 N force acts an angle of θ ° above the positive x-axis. The 7 N force acts at an angle of 70° below the negative x-axis (see diagram).
The resultant of the two forces acts only in the positive x-direction.
Calculate the value of θ.
[2]
Calculate the magnitude of the resultant of the two forces.
[2]
Find the dimensions of energy.
[1]
[1]
[2]
[4]
Using the formula conjectured by the student, determine the value of k.
[3]
The diagram shows a uniform beam AB, of weight 80 N and length 7 m, resting in equilibrium in a vertical plane. The end A is in contact with a rough vertical wall, and the angle between the beam and the upward vertical is 60°. The beam is supported by a smooth peg at a point C, where AC = 2 m.
Complete the diagram in the Printed Answer Booklet to show all the forces acting on the
beam.
[2]
Show that the magnitude of the frictional force exerted on the beam by the wall is 25 N.
[3]
Hence determine the magnitude of the total contact force exerted on the beam by the
wall.
[3]
Determine the direction of the total contact force exerted on the beam by the wall.
[2]
The coefficient of friction between the beam and the wall is μ.
Find the range of possible values for μ.
[2]
Explain how your answer to part (b)(ii) would change if the peg were situated closer to A but
the angle between the beam and the upward vertical remained at 60°.
[1]
Three spheres A, B, and C, of equal radius are in the same straight line on a smooth horizontal surface. The masses of A, B and C are 2 kg, 4 kg and 1 kg respectively.
Initially the three spheres are at rest.
Spheres A and C are each given impulses so that A moves towards B with speed 5 m s-1 and C moves towards B with speed u m s-1 as shown in the diagram below.
The coefficient of restitution between A and B is 5 
It is given that the first collision occurs between A and B.
State how you can tell from the information given above that kinetic energy is lost when A
collides with B.
[1]
Show that the combined kinetic energy of A and B decreases by 24% during their collision.
[5]
Sphere B next collides with C. The coefficient of restitution between B and C is 3 
Given that a third collision occurs, determine the range of possible values for u.
[6]
State one limitation of the model used in this question.
[1]
In the diagram below, points A, B and C lie in the same vertical plane. The slope AB is inclined at an angle of 30° to the horizontal and AB = 5 m. The point B is a vertical distance of 6.5 m above horizontal ground. The point C lies on the horizontal ground.
Starting at A, a particle P, of mass m kg, moves along the slope towards B, under the action of a constant force F. The force F has a magnitude of 50 N and acts at an angle of θ ° to AB in the same vertical plane as A and B. When P reaches B, F is removed, and P moves under gravity landing at C.
It is given that
• the speed of P at A is 3 m s-1,
• the speed of P at B is 6 m s-1,
• the speed of P at C is 12 m s-1,
• 58 J of work is done against non-gravitational resistances as P moves from A to B,
• 42 J of work is done against non-gravitational resistances as P moves from B to C.
By considering the motion from B to C, show that m = 4.33 correct to 3 significant figures.
[3]
By considering the motion from A to B, determine the value of θ.
[4]
Calculate the power of F at the instant that P reaches B.
[2]
A uniform lamina OABC is in the shape of a trapezium where O is the origin of the coordinate system in which the points A, B and C have coordinates (12, 0), (12 + p, q) and (0, q) respectively.
Determine, in terms of p and q, the coordinates of the centre of mass of OABC.
[4]
The point D has coordinates (7.6, q). When OABC is suspended from D, the lamina hangs in equilibrium with BC horizontal.
Determine the value of p.
[3]
When OABC is suspended from C, the lamina hangs in equilibrium with BC at an angle of 35° to the downward vertical.
Determine the value of q, giving your answer correct to 3 significant figures.
[3]
The paper as printed
Spotted a mistake on this page?Report a problem