Simple harmonic motion

A particle P of mass 0.4 kg is moving in a straight line with simple harmonic motion. The maximum speed of P is 5 m s^–1 and the maximum magnitude of acceleration of P is 12.5 m s^–2 The time taken for one complete oscillation is T seconds.

Find the time within each complete oscillation for which P is more than 0.5 m from the centre of the oscillation.

Find the kinetic energy of P at the instant when P is 0.2 m from the centre of the oscillation.

A particle P moves in a straight line with simple harmonic motion about a fixed point O. The magnitude of the greatest acceleration of P is 18 m s^–2

When P is 0.3 m from O, the speed of P is 2.4 m s^–1

The amplitude of the motion is a metres.

Show thata = 0.5

Find the greatest speed of P.

During one oscillation, the speed of P is at least 2 m s^–1 for S seconds.

Find the value of S.

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

The motion of a pendulum, shown in Figure 3, is modelled by the differential equation

where θ is the angle, in radians, that the pendulum makes with the downward vertical, t seconds after it begins to move.

Show that a particular solution of the differential equation is

Hence, find the general solution of the differential equation.

Initially, the pendulum

• makes an angle of radians with the downward vertical

• is at rest

Given that, 10 seconds after it begins to move, the pendulum makes an angle of α radians with the downward vertical,

determine, according to the model, the value of α to 3 significant figures.

Given that the true value of α is 0.62

evaluate the model.

The differential equation

models the motion of the pendulum as moving with forced harmonic motion.

Refine the differential equation so that the motion of the pendulum is simple harmonic motion.

The motion of a pendulum, shown in Figure 3, is modelled by the differential equation

where θ is the angle, in radians, that the pendulum makes with the downward vertical, t seconds after it begins to move.

Show that a particular solution of the differential equation is

Hence, find the general solution of the differential equation.

Initially, the pendulum

• makes an angle of radians with the downward vertical

• is at rest

Given that, 10 seconds after it begins to move, the pendulum makes an angle of α radians with the downward vertical,

determine, according to the model, the value of α to 3 significant figures.

Given that the true value of α is 0.62

evaluate the model.

The differential equation

models the motion of the pendulum as moving with forced harmonic motion.

Refine the differential equation so that the motion of the pendulum is simple harmonic motion.

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 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 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.

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.