Physics_and_Reality_Week_2

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1 Homework Questions for Physics and Reality Week 2 Due the week of February 20 Metric System Prefixes Table T “terra” 10 12 G “giga” 10 9 M “mega” 10 6 k “kilo” 10 3 c “centi” 10 -2 m “milli” 10 -3 μ “micro” 10 -6 n “nano” 10 -9 p “pico” 10 -12 Useful Values: Gravitational constant, G = 6.7 × 10 -11 m 3 kg -1 s -2 Speed of light in a vacuum, c = 3.0 × 10 8 m/s Radius of the Sun, R S = 7.0 × 10 8 m Mass of the Sun, M S = 2.0 × 10 30 kg Radius of Earth, R E = 6.4 × 10 6 m Mass of Earth, M E = 6.0 × 10 24 kg 1 parsec = 1 pc = 3.1 × 10 16 m = 3.1 × 10 13 km 1. Gravity and Black Holes A. In your own words, describe why Einstein was not satisfied with Newton's formulation of gravity (1-2 sentences). B. Which of the following situation pairs are consistent with the equivalence principle? Pick all that apply: a. you sitting on your desk on Earth; you inside a rocket moving at an acceleration of 9.8 m/s 2 b. Einstein experiencing his happiest thought; you experiencing your happiest thought c. you standing on Earth; you standing on the surface of the moon d. a train moving at a constant speed; a rocket at liftoff C. The radius of the event horizon of a black hole is called the Schwarzschild radius (r s ), and as you saw in lecture, it is given by the following equation: Name : Bryan Torres Did you attend lecture?
2 ࠵? ! = 2࠵?࠵? ࠵? " where M is the mass of the black hole, G is the gravitational constant and c is the speed of light. While it is unlikely, a few astronomers have suggested the possibility that there exists a black hole beyond the orbit of Neptune that is still within our solar system, referred to as a Trans-Neptunian Object (TNO), with an approximate mass of 5 times that of Earth. (i) What would the radius of such a black hole be? (ii) Name an object (or place) that has approximately the size of the black hole you calculated in part (i). (iii) In a black hole, all of the mass eventually reaches the infinitely dense central point, or ‘singularity’. However, it can be interesting to calculate an average ‘density’ for a black hole, whereby the black hole’s volume is defined as the volume of a sphere with a radius equal to the Schwarzschild radius. Using this definition, compare the average density of the black hole from part (i) with the density of the Earth. Report your answer in terms of the orders of magnitude difference. 2. Gravity and the Warping of Time As you learned in lecture, time slows down near black holes. In fact, time slows down near all massive objects. Let’s explore those effects. A. Professor Greene is approaching the event horizon of a black hole, while you sit safely in a spaceship far away. Which of the following statements is true?
3 a. You measure that his time is sped up compared to yours, and he measures that your time is sped up compared to his. b. You measure that his time is slowed down compared to yours, and he measures that your time is sped up compared to his. c. You measure that his time is sped up compared to yours, and he measures that your time is slowed down compared to his. d. You measure that his time is slowed down compared to yours, and he measures that your time is slowed down compared to his. B. Now, i magine you and Professor Greene are floating in space, safely outside of the event horizon of a black hole. If you were to suddenly push Prof. Greene towards the event horizon, you would see him appear dimmer and dimmer until he completely disappears from the view when he is at the event horizon. (i) What would you observe about the passage of time for Prof. Greene as he moves towards the event horizon? a. Professor Greene will experience less time for each unit of your time. Eventually he will appear to “freeze” when he reaches the event horizon. b. Professor Greene will appear to “fast forward” through time, aging faster and faster. c. Professor Greene will age at the same rate as you age, but his watch will run slow relative to your watch. d. Professor Greene will age at the same rate as you age, but his watch will run fast relative to your watch. (ii) Instead, suppose that Professor Greene suddenly pushes YOU toward the event horizon of the black hole. What do you observe as you get closer and closer to the event horizon? a. You will observe Professor Greene to age at the same rate as you age, until you cross into the event horizon. At this point, Professor Greene will become suddenly invisible because no light can enter a black hole. b. You will witness Professor Greene experience less time for each unit of your time. Eventually he will appear to “freeze” when you reach the event horizon. c. You will witness Professor Greene age very rapidly and die. You will watch the distant future of the “outside” universe rapidly unfold before your eyes. d. You will observe Professor Greene to age at the same rate as you age, but you will feel your own time pass increasingly slowly.
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4 Consider the following formula, which relates the rate ( T ) at which time passes near any round massive object (not just a black hole) to the rate at which time passes in empty space ( t ): where, once again, G is the gravitational constant, M is the mass of the massive object, and r is the distance to the center of the massive object. C. Recall the formula for the Schwarzschild radius (r s ). Using this formula and the formula above, calculate the rate of passage of time at a distance equal to the Schwarzschild radius (r = r s ) , relative to the passage of time in empty space. D. Maria is in spaceship A, in empty space, very, very far from a black hole, and Norman is in spaceship B close to the same black hole. Maria starts taking the FroSci midterm exam and finishes in 98 minutes, based on her clock. If Norman is at 1.5 Schwarzschild radii away from the center of the black hole, how long according to his clock will the exam have taken Maria?
5 3. Galaxies and Hubble As you heard in lecture, Edwin Hubble discovered that the Universe is expanding. I n the figure below, from Kirshner (2004), we see the measurement of recession velocity (in units km/s) versus distance (in units of megaparsecs, abbreviated Mpc), where each each circle represents the measurement for a single galaxy, derived from Type Ia Supernovae explosions. By measuring the slope of data similar to those shown here, Hubble was able to provide an estimate for the age of the universe. A. In order for us to repeat Hubble’s calculation, we need to first calculate the slope of the line indicated in this figure. In units of km/s per Mpc, calculate the approximate slope of the line that these galaxies create. B . Convert the slope found in part A to units of 1/s by converting the distance in Mpc to distance in km.
6 C. Lastly, the age of the universe can now be estimated by find the inverse of the quantity calculated in part B. Using your previous answer, estimate the age of the universe in Gyr (giga-years). 4. Gravitational Waves This week’s Frontiers video, “Journey of a Gravitational Wave,” is about the LIGO gravitational wave observatory. A. (i) What is a gravitational wave? What is the speed of a gravitational wave? (ii) List one astonishing fact about the gravitational wave event described in the video.
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7 B. (i) Describe in 2-3 sentences how the LIGO observatory is able to detect gravitational waves. (ii) Name one remarkable thing about LIGO.

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