Second order differential equations

Find the extension of each string when the system is in equilibrium.

The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards C, and released from rest.

Show that during the subsequent motion the particle satisfies the differential equation

where x metres is the displacement of the particle below the equilibrium position at time t seconds after the particle is released.

Find x in terms of t

State one limitation of the model used in part (b)

Solve the differential equation

given that y = and = 0 when x = 0

The solution of a second order differential equation is f (t)

The differential equation models heavy damping.

Which one of the statements below could be true?

Tick () one box.

Find the general solution of the differential equation

A bungee jumper of mass m kg is attached to an elastic rope. The other end of the rope is attached to a fixed point.

The bungee jumper falls vertically from the fixed point.

At time t seconds after the rope first becomes taut, the extension of the rope is

A model for the motion while the rope remains taut assumes that the forces acting on the bungee jumper are

the weight of the bungee jumper

a tension in the rope of magnitude kx newtons

an air resistance force of magnitude Rv newtons

where k and R are constants such that 4km R²

Show that this model gives the result

You do not need to find the value of A or the value of B

A second, simpler model assumes that the air resistance is zero.

The values of k, m and g remain the same.

Find an expression for x in terms of t according to this simpler model, giving the values of all constants to two significant figures.

In this question use as 10 m s²

A smooth plane is inclined at 30 to the horizontal. The fixed points A and B are 3.6 metres apart on the line of greatest slope of the plane, with A higher than B

A particle P of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points A and B respectively.

The particle P moves on a straight line that passes through A and B

The natural length of the string AP is 1.4 metres. When the extension of the string AP is e_A metres, the tension in the string AP is

The natural length of the string BP is 1 metre. When the extension of the string BP is e_B metres, the tension in the string BP is

The particle P is held at the point between A and B which is 0.2 metres from its equilibrium position and lower than its equilibrium position. The particle P is then released from rest.

At time t seconds after P is released, its displacement towards B from its equilibrium position is x metres.

Show that during the subsequent motion the object satisfies the equation

Fully justify your answer.

The experiment is repeated in a large tank of oil. During the motion the oil causes a resistive force of kv newtons to act on the particle,

The oil causes critical damping to occur.

Find x in terms of t, giving your answer in exact form.

Calculate the maximum speed of the particle.

On an isolated island some rabbits have been accidently introduced.

In order to eliminate them, conservationists have introduced some birds of prey.

At time t years (t 0) there are x rabbits and y birds of prey.

At time t 0 there are 1755 rabbits and 30 birds of prey.

When t 0 it is assumed that:

the rabbits will reproduce at a rate of a% per year each bird of prey will kill, on average, b rabbits per year the death rate of the birds of prey is c birds per year the number of birds of prey will increase at a rate of d% of the rabbit population per year.

This system is represented by the coupled differential equations:

State the value of a, the value of b, the value of c and the value of d

Solve the coupled differential equations to find both x and y in terms of t

Given that x and y are both positive for 0 t 5, use your answer to part (b) to show that the conservationists’ plan will succeed.

In this question use = 9.8 m s²

A particle P of mass m is attached to two light elastic strings, AP and BP.

The other ends of the strings, A and B, are attached to fixed points which are 4 metres apart on a rough horizontal surface at the bottom of a container.

The coefficient of friction between P and the surface is 0.68

When the extension of string AP is e_A metres, the tension in AP is 24me_A

When the extension of string BP is e_B metres, the tension in BP is 10me_B

The natural length of string AP is 1 metre

The natural length of string BP is 1.3 metres

Show that when AP 1.5 metres, the tension in AP is equal to the tension in BP.

P is held at the point between A and B where AP 1.9 metres, and then released from rest.

At time t seconds after P is released, AP (1.5 x) metres.

Show that when P is moving towards A,

In this question use = 10 m s²

A light spring is attached to the base of a long tube and has a mass m attached to the other end, as shown in the diagram.

The tube is filled with oil.

When the compression of the spring is metres, the thrust in the spring is

The mass is held at rest in a position where the compression of the spring

The mass is then released from rest. During the subsequent motion the oil causes a resistive force of 6mv newtons to act on the mass, where v m s¹ is the speed of the mass.

At time t seconds after the mass is released, the displacement of the mass above its starting position is x metres.

Find x in terms of t.

State, giving a reason, the type of damping which occurs.

Two tanks, A and B, each have a capacity of 800 litres.

At time t 0 both tanks are full of pure water.

When t 0, water flows in the following ways:

Water with a salt concentration of grams per litre flows into tank A at a constant rate Water flows from tank A to tank B at a rate of 16 litres per minute Water flows from tank B to tank A at a rate of r litres per minute Water flows out of tank B through a waste pipe The amount of water in each tank remains at 800 litres.

At time t minutes (t 0) there are x grams of salt in tank A and y grams of salt in tank B.

This system is represented by the coupled differential equations

Find the value of r.

Show that 3

Solve the coupled differential equations to find both x and y in terms of t.