Further Mechanics

Question 1

What is the equation for Hooke’s law?

Answer 1

T = λx / l

Question 2

What is the equation for energy stored in a spring?

Answer 2

E = λx² / 2l

Question 3

What are the units of modulus of elasticity?

Answer 3

N

Question 4

What is the equation for the final velocity and angle of a particle with initial velocity U which collides with a wall at angle α, and coefficient of restitution e.

Answer 4

Vcosβ = Ucosα
Vsinβ = eUsinα

⇒ tanβ = etanα
⇒ V = U√(e²sin²α + cos²α)

Question 5

How can you work out the angle of deflection?

Answer 5

cosθ = (U.V) / (|U||V|)

Question 6

What are sin(90-x), cos(90-x) and tan(90-x)?

Answer 6

sin(90-x) = cosx
cos(90-x) = sinx
tan(90-x) = cotx

Question 7

What’s the strategy for just finding the magnitude of an impulse in a collision with a wall?

Answer 7

• Just look at the perpendicular components of velocity before and after.
• Remember to multiply by mass.
• You can do mUsinα (e+1)

Question 8

What’s the strategy for impacts on a wall of an unknown angle?

Answer 8

• Find a unit vector Î perpendicular to the wall, in the direction of the impulse.
• Find a unit vector W^ parallel to the wall, in the general direction of U.
• U = (I.U)I + (W.U)W, and same for V

https://www.youtube.com/watch?v=7iZ2USrIpuE

Question 9

How do you find e given u and v?

Answer 9

e = ∆v.v / ∆v.u

Question 10

What do you always get wrong about the line of centres?

Answer 10

The line of centres is of course the line containing both of the circles’ centres. You always seem to think this is the line tangent to both circles.

Question 11

What should you remember when finding a range of values for e?

Answer 11

By default, 0 ≤ e ≤ 1

Question 12

What are important exam techniques for the mechanics paper?

Answer 12

• Make sure you’ve worked out what you are trying to find before you start working, so you know you’re on the right track.
• Take time to organise your work: label your variables on the diagram (speeds: v, w), and lay everything out neatly so you can retrieve old information faster and correct mistakes more easily.
• Always put a heading above your work with the principle you are using: law of restitution, conservation of momentum (in x, y), work-energy principle.
• Don’t rush, or you end up making silly mistakes, such as wrong direction of velocity, formula written out wrong (a = F/m!!), forgetting to multiply by e, misreading the question, using the length rather than the extension, etc.
• Sense check for silly mistakes, e.g. ∆EPE should always be in terms of A, second velocity should have a smaller magnitude than initial, the angle should be at least similar to what is shown on the diagram.

Question 13

How do you use the assumption that spheres have equal radii?

Answer 13

The line of centres is parallel to the surface the spheres are moving
on, so the impulse acts parallel to the surface.