# 1 The displacement, s in metres, of an object after a time, t in seconds, is given by s = 90t 4 t 2

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1 CFE Advanced Higher Physics Unit 1 Rotational Motion and Astrophysics Kinematic relationships 1 The displacement, s in metres, of an object after a time, t in seconds, is given by s = 90t 4 t 2 a) Find by differentiation the equation for its velocity. b) At what time will the velocity be zero? c) Show that the acceleration is a constant and state its value. 2 3 dv Given that a =, show by integration that the velocity, v, is given by v = u + at. dt State clearly the meaning of the symbol, u, in this equation. ds Given that v = and v = u + at, show by integration that s = ut + ½at 2. dt Where the symbols have their usual meaning. 4 The displacement, s, of a moving object after a time, t, is given by s = 8 10t + t 2. Show that the unbalanced force acting on the object is constant. 5 The displacement, s, of an object after time, t, is given by s = 3t 3 + 5t. a) Derive an expression for the acceleration of the object. b) Explain why this expression indicates that the acceleration is not constant. 6 A trolley is released from the top of a runway which is 6 m long. The displacement, s in metres, of the trolley is given by the expression s = 5t + t 2, where t is in seconds. Determine: a) an expression for the velocity of the trolley b) the acceleration of the trolley c) the time it takes the trolley to reach the bottom of the runway d) the velocity of the trolley at the bottom of the runway. 7 A box slides down a smooth slope with an acceleration of 4 m s -2. The velocity of the box at a time t = 0 is 3m s -1 down the slope. dv Using a = show by integration that the velocity, v, of the box is given by v = 3 + 4t. dt 8 The equation for the velocity, v, of a moving trolley is v = 2 + 6t. ds Using v = derive an expression for the displacement, s, of the trolley. dt Angular Motion 1 Convert the following from degrees to radians: 30 o, 45 o, 60 o, 90 o, 180 o, 270 o, 360 o, 720 o.

7 b) The mass of the neutron star is the same as the mass of the sun. What is the density of the neutron star? c) The radius of a neutron is about m. Estimate the average spacing of the neutrons in the neutron star. Gravitation 1 State the inverse square law of gravitation 2 Calculate the gravitational force between two cars parked 0.50 m apart. The mass of each car is 1000 kg. 3 Calculate the gravitational force between the Earth and the Sun. 4 a) By considering the force on a mass, at the surface of the Earth, state the expression for the gravitational field strength, g, in terms of the mass and radius of the Earth. b) (i) The gravitational field strength is 9.8 N kg -1 at the surface of the Earth. Calculate a value for the mass of the Earth. (ii) Calculate the gravitational field strength at the top of Ben Nevis, 1344 m above the surface of the Earth. (iii) Calculate the gravitational field strength at 200 km above the surface of the Earth. 5 a) What is meant by the gravitational potential at a point? b) State the expression for the gravitational potential at a point c) Calculate the gravitational potential: (i) at the surface of the Earth (ii) 800 km above the surface of the Earth. 6 A gravitational field is a conservative field. Explain what is meant by this statement. 7 Calculate the energy required to place a satellite of mass kg into an orbit at a height of 350 km above the surface of the Earth. 8 Which of the following are vector quantities: gravitational field strength gravitational potential escape velocity, universal constant of gravitation gravitational potential energy period of an orbit 9 A mass of 8.0 kg is moved from a point in a gravitational field where the potential is 15 J kg -1 to a point where the potential is 10 J kg -1. a) What is the potential difference between the two points? b) Calculate the change in potential energy of the mass. c) How much work would have to be done against gravity to move the mass between these two points? 10 a) Explain what is meant by the term escape velocity. b) Derive an expression for the escape velocity in terms of the mass and radius of a planet. c) (i) Calculate the escape velocity from both the Earth and from the Moon. (ii) Using your answers to (i) comment on the atmosphere of the Earth and the Moon.

8 11 Calculate the gravitational potential energy and the kinetic energy of a 2000 kg satellite in geostationary orbit above the Earth. 12 a) A central force required to keep a satellite in orbit. Derive the expression for the orbital period in terms of the orbital radius. b) A satellite is placed in a parking orbit above the equator of the Earth. (i) State the period of the orbit. (ii) Calculate the height of the satellite above the equator. (iii) Determine the linear speed of the satellite (iv) Find the central acceleration of the satellite 13 A white dwarf star has a radius of 8000 km and a mass of 1.2 solar masses a) Calculate the density of the star in kg m -3. b) Find the gravitational potential at a point on the surface. c) Calculate the acceleration due to gravity at a point on the surface d) Estimate the potential energy required to raise your centre of gravity from a sitting position to a standing position on this star. e) A 2 kg mass is dropped from a height of 100 m on this star. How long does it take to reach the surface of the star? 14 a) How are photons affected by a massive object such as the Sun? b) Explain, using a sketch, why light from a distant star passing close to the Sun may suggest that the star is at a different position from its true position. c) Explain what is meant by the term black hole. General Relativity 1 What is meant by an inertial frame of reference? 2 What is meant by a non-inertial frame of reference? 3 Which frame of reference applies to the theory of special relativity (studied in Higher Physics)? 4 At what range of speeds do the results obtained by the theory of special relativity agree with those of Newtonian mechanics? 5 How does the equivalence principle link the effects of gravity with acceleration? 6 In which part of an accelerating spacecraft does time pass more slowly? 7 Does time pass more quickly or more slowly at high altitude in a gravitational field? 8 How many dimensions are normally associated with space-time?

9 9 Two space-time diagrams are shown, with a worldline on each. Write down what each of the worldliness describes. a) t b) t x x 10 The space-time diagram shows two worldlines. Which worldline describes a faster speed? (These speeds are much less than the speed of light.) t 11 Explain the difference between these two worldlines on the space -time diagram. x t 12 Explain what is meant by the term geodesic. 13 Copy the following space-time diagram. x t x Insert the following labels onto your diagram: the present the future the past v = c v c v c. 14 What effect does mass have on spacetime? 15 Describe two situations where a human body experiences the sensation of force. 16 How does general relativity interpret the cause of gravity? 17 Mercury s orbit around the Sun could not be predicted accurately using classical mechanics. General relativity was able to predict Mercury s orbit accurately. Investigate this using a suitable search engine and write a short paragraph summarising your results.

10 18 A star of mass kg collapses to form a black hole. Calculate the Schwarzschild radius of this black hole. 19 A star of mass equivalent to six solar masses collapses to form a black hole. Calculate the Schwarzschild radius of this black hole. 20 If our Sun collapsed to form a black hole, what would be the Schwarzschild radius of this black hole? 21 If our Earth collapsed to form a black hole, what would be the Schwarzschild radius of this black hole? 22 A star is approximately the same size as our Sun and has an average density of kg m 3. If this star collapsed to form a black hole, calculate the Schwarzschild radius of the black hole. Stellar physics 1 A star emits electromagnetic radiation with a peak wavelength of m. a) Use Wien s law (λ max T = ) to calculate the surface temperature of the star. b) Calculate the power of the radiation emitted by each square metre of the star s surface where the star is assumed to be a black body. Stefan Boltzmann constant = J s 1 m 2 K 4. 2 The Sun has a radius of m and a surface temperature of 5800 K. a) Calculate the power emitted per m 2 from the Sun s surface. b) Calculate the luminosity of the Sun. c) Calculate the apparent brightness of the Sun as seen from the Earth. 3 Three measurements of a distant star are possible from Earth. These measurements are: apparent brightness = W m 2 peak emitted wavelength = m distance to star (parallax method) = m a) Use Wien s law (λ max T = ) to calculate the surface temperature of the star. b) Calculate the energy emitted by each square metre of the star s surface per second. c) Calculate the luminosity of the star. d) Calculate the radius of the star. 4 A star is 86 ly from Earth and has a luminosity of W m 2. Calculate the apparent brightness of the star. 5 The apparent brightness of a star is W m 2. The star is 16 ly from Earth. Calculate the luminosity of the star. 6 A star with luminosity W m 2 has an apparent brightness of W m 2 when viewed from Earth. Calculate the distance of the star from Earth: a) in metres

11 b) in light years. 7 A star with radius m and surface temperature 6300 K has an apparent brightness of W m 2. Calculate its distance from the Earth. 8 A star with radius m and surface temperature 5900 K is 36 ly from Earth. Calculate the apparent brightness of the star. 9 Show mathematically that the luminosity of a star varies directly with the square of its radius and the fourth power of its surface temperature. 10 Show mathematically that the apparent brightness of a star varies directly with the square of its radius and the fourth power of its surface temperature and varies inversely with the square of its distance from the Earth. 11 Two stars, A and B, are the same distance from the Earth. The apparent brightness of star A is W m 2 and the apparent brightness of star B is W m 2. Show that star A has 20 times the luminosity of star B. 12 A star has half of our Sun s surface temperature and 400 times our Sun s luminosity. How many times bigger is the radius of this star compared to the Sun? 13 Information about two stars A and B is given below. surface temperature of star A = 3 surface temperature of star B radius of star A = 2 radius of star B a) How many times is the luminosity of star A greater than the luminosity of star B? b) Stars A and B have the same apparent brightness from Earth. Which star is furthest from Earth and by how many times?

12 14 The diagram shows one way of classifying stars. Each dot on the diagram represents a star. units a) What name is usually given to this type of diagram? b) The stars are arranged into four main regions. Identify the region called: i) the main sequence ii) giants iii) super giants iv) white dwarfs. c) i) In which of the regions on the diagram is the Sun? ii) The surface temperature of the Sun is approximately 5800 K. Explain why the scale on the temperature axis makes it difficult to identify which dot represents the Sun. d) In which region would you find the following: i) a hot bright star ii) a hot dim star iii) a cool bright star iv) a cool dim star? e) A star is cooler than, but brighter than the Sun. i) What can be deduced about the size of this star compared to the size of the Sun? ii) What region would this star be in? f) A star is hotter than, but dimmer than, the Sun. i) What can be deduced about the size of this star compared to the size of the Sun? ii) What region would this star be in? g) The Sun s nuclear fuel will be used up with time. What will then happen to the Sun s position in the above diagram?

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