Elastic Strings & Springs (Edexcel A Level Further Maths: Further Mechanics 1): Exam Questions

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 .

The point B is vertically below O such that .

The particle is held at rest at A, then released and first comes to instantaneous rest at the point B.

Show that .

 Find, in terms of g, the acceleration of P immediately after it is released from rest at A.

Find, in terms of g and a, the maximum speed attained by P as it moves from A to B.

A light elastic string with natural length and modulus of elasticity  has one end attached to a fixed point on a rough  inclined plane. The other end of the string is attached to a package of mass .

The plane is inclined at an angle  to the horizontal, where .

The package is initially held at . The package is then projected with speed up a line of greatest slope of the plane and first comes to rest at the point , where . 

The coefficient of friction between the package and the plane is .

By modelling the package as a particle,

show that .

Find the acceleration of the package at the instant it starts to move back down the plane from the point .

Figure 2

A light elastic spring has natural length and modulus of elasticity .

One end of the spring is attached to a fixed point on a rough inclined plane.

The other end of the spring is attached to a package of mass . 

The plane is inclined to the horizontal at an angle  where . 

The package is initially held at the point  on the plane, where . The point  is higher than  and is a line of greatest slope of the plane, as shown in Figure 2. 

The coefficient of friction between  and the plane is .

By modelling  as a particle,

show that the acceleration of  at the instant when is released from rest is .

Find, in terms of and , the speed of  at the instant when the spring first reaches its natural length of .

A spring of natural length has one end attached to a fixed point . The other end of the spring is attached to a package of mass .
The package is held at rest at the point , which is vertically below such that .
After being released from rest at , the package first comes to instantaneous rest at .
Air resistance is modelled as being negligible.

By modelling the spring as being light and modelling as a particle, show that the modulus of elasticity of the spring is .

(i) Show that attains its maximum speed when the extension of the spring is .

(ii) Use the principle of conservation of mechanical energy to find the maximum speed, giving your answer in terms of and .

In reality, the spring is not light.

State one way in which this would affect your energy equation in part (b).

A light elastic string has natural length and modulus of elasticity .
One end of the elastic string is attached to a fixed point . A particle of mass is attached to the other end of the elastic string.
The particle hangs freely in equilibrium at the point , which is vertically below

Find the length .

Particle is now pulled vertically downwards to the point , where , and released from rest. The resistance to the motion of is a constant force of magnitude .

Find, in terms of and , the speed of after it has moved a distance .

Particle is now held at O.
Particle is released from rest and reaches its maximum speed at the point .

The resistance to the motion of is again a constant force of magnitude .

Find the distance .

A light elastic string has natural length 2 and modulus of elasticity 2.
One end of the string is attached to a fixed point on a horizontal ceiling.
The other end is attached to a particle of mass .

The particle hangs in equilibrium at the point , where .

The particle is then projected vertically downwards from with speed

Air resistance is assumed to be negligible.

Find the elastic energy stored in the string, when first comes to instantaneous rest.
Give your answer in the form , where is a constant to be found.