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Name: _______________________________________________Edexcel A Level Further Mathematics : Further Mechanics
Elastic strings and springs and elastic energy
Date:
Time:
Total marks available:
Total marks achieved: ______
Questions
,Q1.
A spring of natural length a has one end attached to a fixed point A. The other end of the spring
is attached to a package P of mass m.
The package P is held at rest at the point B, which is vertically below A such that AB = 3a.
After being released from rest at B, the package P first comes to instantaneous rest at A.
Air resistance is modelled as being negligible.
By modelling the spring as being light and modelling P as a particle,
(a) show that the modulus of elasticity of the spring is 2mg
(5)
(b) (i) Show that P attains its maximum speed when the extension of the spring is
a
(ii) Use the principle of conservation of mechanical energy to find the maximum speed, giving
your answer in terms of a and g.
(6)
In reality, the spring is not light.
(c) State one way in which this would affect your energy equation in part (b).
(1)
(Total for question = 12 marks)
Q2.
Figure 5 represents the plan view of part of a smooth horizontal floor, where RS and ST are
smooth fixed vertical walls. The vector is in the direction of i and the vector
, is in the
direction of (2i + j).
A small ball B is projected across the floor towards RS. Immediately before the impact with RS,
the velocity of B is (6i – 8j) m s–1. The ball bounces off RS and then hits ST.
The ball is modelled as a particle.
Given that the coefficient of restitution between B and RS is e,
(a) find the full range of possible values of e.
(3)
It is now given that e =
and that the coefficient of restitution between B and ST is
(b) Find, in terms of i and j, the velocity of B immediately after its impact with ST.
(7)
(Total for question = 10 marks)
Q3.
A particle P, of mass m, is attached to one end of a light elastic spring of natural length a
and modulus of elasticity kmg.
The other end of the spring is attached to a fixed point O on a ceiling.
The point A is vertically below O such that OA = 3a
The pointB is vertically below O such that OB =
a
The particle is held at rest at A, then released and first comes to instantaneous rest at the point
B.
(a) Show that k =
(3)
(b) Find, in terms of g, the acceleration of P immediately after it is released from rest at A.
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