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18.2 Newton’s law of gravitation Stretch and challenge OCR Physics A Finding the mass of the Earth, the Sun, and the black hole at the centre of our galaxy Specification references 5.2.2c, 5.4.2a, 5.4.3c M 0.1, 0.2, 2.2, 2.3 Introduction In this activity you will learn how we measure the mass of large objects: the Earth, the Sun, and finally the super-massive black hole at the centre of our galaxy, the Milky Way. Some of these measurements were made centuries ago in the time of Newton but the measurements on black holes and confirmation of their existence are still continuing. You will also apply the idea of escape velocity to a black hole. Where an object approaches the speed of light then the equations of relativity should really be used to obtain exact answers. In these questions you will use Newton’s equations but the answers are surprisingly accurate! Learning outcomes After completing the worksheet you should be able to: understand and apply Newton’s law of gravitation calculate centripetal acceleration. Background Mass of the Earth: All you need to find the mass of the Earth is the acceleration due to gravity at the Earth’s surface, 9.81 m s−2, the radius of the Earth, 6.4 × 106 m, and, of course, Newton’s law of gravitation. Mass of the Sun: We cannot measure the acceleration due to gravity on the surface of the Sun in the same way as we can on the surface of the Earth. To find the mass of the Sun we need to know the period of a planet in orbit around the Sun and its distance from the Sun. The Earth is 146 million kilometres from the Sun and orbits the Sun in 1 year. © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 1 18.2 Newton’s law of gravitation Stretch and challenge OCR Physics A Mass of a black hole: The article below has been adapted from www.space.com. Surprising observations of a star swiftly orbiting the cloudy heart of the Milky Way Galaxy have verified with near certainty the existence of a central black hole, a theoretical object that still eludes direct detection. Astronomers watched the star for a decade, tracking two-thirds of its path around the galactic centre. No object has ever before been seen so close to the centre of any galaxy, nor has any other object previously been observed making more than a small fraction of its orbital trek around a galaxy. "Our work proves that there is indeed a super massive black hole in our own galaxy," said Rainer Schoedel, a PhD student at the Max-Planck Institute for Extraterrestrial Physics (MPE) in Germany. An international team of astronomers photographed the star as it zoomed around the galactic centre at speeds ultimately exceeding 11 million mph (5,000 kilometres per second). Early this year, the star flitted precariously close to the black hole, coming within 17 light-hours, or just three times the distance from the Sun to Pluto. The observations rule out nearly all other possible explanations for the tremendous amount of matter – equal to some 2.6 million Suns – packed into a tight spot at the centre of our galaxy. © www.space.com Questions You will need to use G = 6.67 ´10-11 Nm2 kg-2 and c = 3.00 ´108 ms-1 1 Find the mass of the Earth. (2 marks) 2 Find the mass of the Sun. (2 marks) 3 a i Use data from the article to calculate the closest distance from the star to the black hole. (2 marks) © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 2 18.2 Newton’s law of gravitation Stretch and challenge OCR Physics A ii Calculate the centripetal acceleration of the star moving at 5000 km s−1 at the closest distance of approach. (2 marks) iii Assuming that the star is in a circular orbit around the black hole, estimate the mass of the black hole. (3 marks) iv Hence, show that the mass of the black hole, some 2.6 million Suns, as stated in the article, is a reasonable estimate. (2 marks) v Apart from using equations that do not take into account relativity, state one reason why the two values do not completely agree. (2 marks) b As gas, dust, and sometimes even stars fall into a black hole, electromagnetic waves are emitted. Radio waves emitted from the black hole at the centre of the Milky Way appear to come from an object about 1 1010 m in radius. It is not possible to escape if an object comes too close to a black hole – within what is called the event horizon. i Write down the formula for the work done to remove an object of mass, m, to infinity if it is a distance R from an object of mass M. (1 mark) © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 3 18.2 Newton’s law of gravitation Stretch and challenge OCR Physics A ii If the object of mass m has enough kinetic energy it can escape the black hole. Estimate the escape velocity for an object 1 × 1010 m away from the black hole at the centre of our galaxy. Use the mass of the black hole calculated in a iii. (3 marks) iii Comment on the value you obtained in ii. (2 marks) © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 4