UNIVERSITY OF CAMBRIDGE UNINTENTIONAL EXAMINATIONS

General Certificate of Education

Advanced Subsidiary Level and Advanced Level

FURTHER MATHEMATICS A2 9231/02

Paper 2 Mechanics November 2008

Up to 1.5 hours

Additional Materials: Answer Booklet/Paper

READ THESE INSTRUCTIONS FIRST

If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your English Name on all the work you hand in.

Write in dark blue or black pen.

You may use a soft pencil for any diagrams or graphs.

Do not use staples, paper clips, highlighters, glue or correction fluid (all of these are excluded examination materials).

Make sure your phone is turned off. Make sure anything that could be described as notes is well out of reach.

Put any drink out of sight and leave it there. Check that you are wearing your uniform properly.

Answer all the questions.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.

The use of an electronic calculator is expected, where appropriate. Formulae are assumed to be learned.

You are reminded of the need for clear presentation in your answers.

The value of g is 10 m/s/s unless you are told differently in a question.

At the end of the test, hand in ALL the papers you received; both question paper and your worked answers.

Make sure your English name is on every answer sheet.

The number of marks is given in brackets [ ] at the end of each question or part question.

The total number of marks for this paper is 60.

This document consists of 2 printed pages

[Turn over]

1. Draw an appropriate diagram and write an expression for:

i)the work done when a box of mass M is dragged distance D up a slope at angle ø to the horizontal against a frictional force F by a force P applied parallel to the plane through the centre of mass of the box.

ii) The exact moment applied by a constant force F applied at one end of, and at 60º to the line of, a 1200mm rod. The other end is fixed to a spindle, whose axis lies perpendicular to the plane created by of the rod and force F.

iii)The impulse of the force F applied by a tennis racquet when it hits a 58g ball travelling at 20m/s and returns it at 25 m/s

iv)The position of the centre of mass of a solid regular circular cone

v)The position of the centre of mass of an isosceles triangle with sides 10, 13, 13 cm, formed from a uniform wire.[10]

2. A tennis ball is hit at up to 60 m/s by an experienced player. Ignoring air resistance throughout this question and assuming g = 9.8 exactly,

i) How far would it be possible to send the ball before it bounces for the first time?

ii) How high could the ball be hit?

iii) In what direction would you hit the ball so that it stayed in the air for the longest time? Justify this answer.[2,2,2]

A ball in play arrives at its first bounce at 25º to the horizontal, travelling at 30 m/s. The coefficient of restitution of a tennis ball is very close to 1/√2.

iv) What is the velocity of the ball after the bounce?[4]

3. A small weight of W is suspended vertically from a point O by an elastic string ON of natural length l (ON = l ) and stiffness k. The equilibrium position, when the weight is suspended and stationary is at point E below O, and a distance e below the natural length (so NE = e). The weight is lifted so that the string is only just slack and released from rest.

i) Form an equation of motion for the weight and show that it moves in simple harmonic motion. Find the period and amplitude of the motion.[6,2]

ii) What happens to the motion if the weight is released from a point Q below O on ON such that QN = e ? How long is it before the weight returns to Q for the first time?[4]

4. Mass A, of 5kg is travelling on a smooth surface at 6 m/s and meets mass B of 3kg while it is at rest. The coefficient of restitution between the two is 0.6.

i) Show that the subsequent motion results in B travelling at 6 m/s. [5]

B bounces against a wall perpendicular to the line of motion with a coefficient of restitution between the two of 0.5.

ii) Describe the subsequent motion[8]

DJS 20081105

Q1 Diagrams often poor, sometimes pictures. Only first one earns marks. (i) should really be an integration with D as the upper bound. (ii) non-readers added gravity and resistance (iii) ball is this mass - how to get this wrong? (iv) No-one knew it, a few worked it out correctly (v) The result is not the same as the lamina of the same shape; put an axis along the line of symmetry...

Q2 Several ignored the 60m/s. Most didn’t think about the height of hitting the ball (left vague to criticise accuracy). Few knew the formulae, and no-one worked them out correctly. The handling of oblique impact was not grapsed by more than one.

Q3 Proof of SHM read like a text-book with no comprehension - which is why I gave W, not m. Expect an answer with √(e/g) in it. The latter part, where it is ballistic, left everyone adrift.

Q4 The bounces continue, tranferring A’s speed to B, which hits the wall at least twice - hence 8 marks.

Pathetic reception: these kids can’t read either. One candidate turned up 5 mins late, did two marks worth only - and fell asleep. He was hit with the A2 Pure that afternoon, faring better, but not as good as an FM student should !!