Power

Gravitational waves are disturbances in space-time that are produced by violent astrophysical events.

Define strain.

The maximum strain shown on the graph is approximately 10^–21. (ii)

Calculate the force that would produce the same strain in a copper wire of diameter 2 mm.

Young modulus of copper = 130 GPa

By considering the answer to (ii), suggest why it is very difficult to detect the strain caused by gravitational waves.

In this part of the question you will use Newtonian physics to estimate the power output of the black hole collision.

The diagram shows two identical black holes, each of mass M, orbiting each other with period T.

Prior to colliding, the black holes move at speed V in identical circular orbits of diameter D about the central point C.

Write down an expression for the gravitational force, F, acting between the black holes in terms of M, D and the universal gravitational constant, G.

The total orbital energy, E, of the black holes is the sum of their kinetic energy and gravitational potential energy. Show that E is given by the expression:

E = k (²G²M⁵)^1/3 (1/T^2/3)

where k is a numerical constant.

By considering times B and D, use your answer to (iii) to estimate the power radiated as gravitational waves in this time interval.

Describe how to find the centre of mass of a 2-dimensional shape, including any equipment required.

Fig. 19 shows a bridge.

The bridge can be raised by an electric motor to allow tall ships to pass underneath.

The moving section of the bridge is 14 m long and has a weight of 120 kN.

The centre of mass of the structure is 6.3 m from the pivot.

Calculate the average power required to raise the bridge to a vertical position in 90 s.

Suggest why the actual electric motor used to lift the bridge has a maximum power output several times larger than the value calculated in (b)(i).

A pulsar is a rapidly rotating neutron star that emits radio waves.

Describe the formation of a neutron star.

State one characteristic of a neutron star.

An astronomer uses a radio telescope to observe a pulsar.

The graph below shows the power that the telescope receives due to the radio waves from one full rotation of a pulsar.

By calculating the area between the curve and the horizontal axis, estimate the total energy received by the telescope in one full rotation of the pulsar.

The surface area of the telescope is about 3000 m². (ii)

The distance to the pulsar is about 300 pc.

By assuming that the radiation from the pulsar is emitted equally in all directions, estimate the total energy emitted in one full rotation.

Large power stations generate an electrical power of about 1 GW.

Current methods of energy production that use nuclear fusion are unable to produce enough energy for large-scale energy production. A proposed method of controlling nuclear fusion is inertial confinement fusion (ICF). ICF uses a large number of powerful lasers to create the high temperatures required for nuclear fusion to occur.

One ICF experiment uses a network of capacitors to store the energy needed to power the lasers. When the network is fully charged:

• potential difference across the network = 24 kV

• total energy stored in the network = 400 MJ

Calculate the total capacitance, C, of the network.

Explain why the individual capacitors in the network should be connected in parallel in order to produce this total capacitance.

The total stored energy must be released in a time of less than 1 millisecond.

Explain, using a calculation, why the lasers are powered by the network of capacitors instead of being connected directly to the mains electricity supply.

The fusion reaction in the ICF experiment is

Calculate the number of fusion reactions that must occur for the energy released by fusion to be equal to the electrical energy stored in the network of capacitors.

• mass of deuterium = 2.014102 u

• mass of tritium = 3.016049 u

• mass of alpha particle = 4.002603 u

• mass of neutron = 1.008665 u

Explain the shape of the curve in Fig. 3.3.

The torch is inverted 80 times.

Calculate the total energy stored in the capacitor.

A plastic kettle is filled with 0.60 kg of water at a temperature of 20 °C. A 2.2 kW electric heater is used to heat the water for a time of 4.0 minutes.

Calculate the total energy supplied by the heater during the time of 4.0 minutes.

The specific heat capacity of water is 4200 J kg⁻¹ K⁻¹ and the specific latent heat of (b) vaporisation of water is 2.3 × 10⁶ J kg⁻¹. The boiling point of water is 100 °C.

Calculate the mass of water remaining in the kettle after 4.0 minutes. Assume that all the thermal energy from the heater is transferred to the water.