Elastic strings & springs

A smooth plane is inclined at an angle θ to horizontal ground, where sin θ The points A, B and C lie on a line of greatest slope of the plane, with B between A and C, with A above B and AC = 3a, as shown in Figure 3.

A light elastic string of natural length 2a has one end attached to the fixed point A. The other end of the string is attached to a parcel P of mass m. The modulus of elasticity of the string is mg The parcel rests in equilibrium on the plane at the point B.

The parcel is modelled as a particle.

The parcel is now held at C and released from rest.

Find the elastic potential energy lost by the string when P moves from C to B.

Use the principle of conservation of mechanical energy to find the speed of P at the instant it passes B.

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

The particle P hangs in equilibrium at the point E, where AE = 3a. The particle P is then projected vertically downwards from E with speed

Air resistance is assumed to be negligible.

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

A light elastic string has natural length 2 a and modulus of elasticity 4 mg. One end of the elastic string is attached to a fixed point O. A particle P of mass m is attached to the other end of the elastic string. The particle P hangs freely in equilibrium at the point E, which is vertically below O

Find the length OE.

Particle P is now pulled vertically downwards to the point A, where OA = 4a, and released from rest. The resistance to the motion of P is a constant force of

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

Particle P is now held at O Particle P is released from rest and reaches its maximum speed at the point B. The resistance to the motion of P is again a constant force of magnitude mg.

Find the distance OB.

The fixed points A and B lie on a smooth horizontal surface with AB 6m.

A particle P has mass 0 3.kg.

One end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N, is attached to P, and the other end is attached to A.

One end of another light elastic string, of natural length 2 m and modulus of elasticity 40 N, is attached to P and the other end is attached to B.

The particle P is at rest in equilibrium at the point E on the surface, as shown in Figure 7.

The particle P is now held at the midpoint of AB and released from rest.

Show that P oscillates with simple harmonic motion about the point E.

The time between the instant when P is released and the instant when it first returns to the point E is S seconds.

Find the exact value of S.

Find the length of time during one oscillation for which the speed of P is more

Throughout this question, use g = 10 m s^–2

A light elastic string has natural length 1.25 m and modulus of elasticity 25 N.

A particle P of mass 0.5 kg is attached to one end of the string. The other end of the string is attached to a fixed point A. Particle P hangs freely in equilibrium with P vertically below A

The particle is then pulled vertically down to a point B and released from rest.

Show that, while the string is taut, P moves with simple harmonic motion with

The maximum kinetic energy of P during the subsequent motion is 2.5 J.

Show thatAB = 2 m

The particle returns to B for the first time T seconds after it was released from rest at B

Find the value of T

A light elastic spring has natural length 3l and modulus of elasticity 3mg.

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

The other end of the spring is attached to a package P of mass m. The plane is inclined to the horizontal at an angle α where tanα =

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

The package is released from rest at Y and moves up the plane. The coefficient of friction between P and the plane is

By modelling P as a particle,

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

find, in terms of g and l, the speed of P at the instant when the spring first reaches its natural length of 3 l.

A light elastic string, of natural length l and modulus of elasticity 2mg, has one end attached to a fixed point A and the other end attached to a particle P of mass m. The particle P hangs in equilibrium at the point O.

The particle P is pulled down vertically from O to the point B, where OB = l, and released from rest.

Air resistance is modelled as being negligible.

Using the model,

prove that P begins to move with simple harmonic motion about O with

The particle P first comes to instantaneous rest at the point C.

Using the model,

find the length BC in terms of l,

find, in terms of l and g, the exact time it takes P to move directly from B to C.

A light elastic string with natural length l and modulus of elasticity kmg has one end attached to a fixed point A on a rough inclined plane. The other end of the string is attached to a package of mass m. The plane is inclined at an angle θ to the horizontal, where tanθ =

The coefficient of friction between the package and the plane is

By modelling the package as a particle,

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

A light elastic spring has natural length l and modulus of elasticity 4mg. A particle P of mass m is attached to one end of the spring. The other end of the spring is attached to a fixed point A. The point B is vertically below A with AB= . The particle P is released from rest at B.

Find, in terms of m, l and g, the maximum kinetic energy of P during the motion.

Find the time within each complete oscillation for which the length of the spring is less than l.