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

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 .