IB 26.2 - Energy and General Relativity

Get the Formative Assessment: FA26.2.docx

Help with the: Practice Problems

The Formative Assessment Solved:

1. What is the kinetic energy of a 0.142 kg baseball going 0.780c?

(7.64x10¹⁵ J)

2. What is the velocity of a 0.128 MeV electron?

(0.600 c)

3. A black hole has an event horizon of 4.30x10^6 m in radius. What must be its mass?

(2.90x10³³kg)

4. A very strong concertmaster is on top of a 148 m tall tower near a black hole where the gravitational field strength is 5.53x10¹² N/kg. If another player is making a frequency of 440.0 Hz at the bottom, what frequency does the concertmaster hear at the top? What beat frequency do they hear?

(436. Hz, and 4.00 beats per second)

5. A black hole has a mass of 21.0 solar masses. (The mass of the sun is 1.99x10³⁰ kg.) *Calculate the radius of the event horizon, *Calculate the time it would take a clock 160. km beyond the event horizon to register 60.0 seconds of elapsed time as we observe it from a great distance.

(61.9 km, 70.666 s)

Practice Problems 26.2 - Energy and General Relativity

Questions:

Where does the mass come from when it dilates (increases) with velocity. <Video>

Problems:

2. A 1.2 kg object (rest mass) is moving at 2.85x10⁸ m/s. What is its new mass, and what is its kinetic energy? (3.84 kg, 2.4x10¹⁷ J) <Video>

3. What velocity must an electron have when it is accelerated through 340. kV? m_o = 9.11x10^-31 kg = 0.511 MeV (0.800c) <Video>

4. A 0.00612 kg (rest mass) bullet is going so fast it has a (dilated) mass of 0.00645 kg. What is its kinetic energy, and what is its velocity (2.97x10¹³ J, 0.32c) <Video>

5. The LHC will accelerate protons to about 7 TeV. If a proton has a rest mass of 938 MeV, what is the velocity of the protons in the LHC? (0.999999991c) <Video>

6. A 0.146 kg baseball has a kinetic energy of 3.30x10¹⁶ J. What is its moving mass in kg? What is its velocity? (0.513 kg, 0.959c) <Video>

7. An electron is going 0.980 c. What is its kinetic energy in MeV? (2.06 MeV) <Video>

8. A 0.160 kg baseball is going so fast that it has a mass of 0.190 kg. How fast is it going? What is its kinetic energy in Joules? (0.539c, 2.07x10¹⁵ J) <Video>

9. A proton is accelerated though 210. million volts. What is its velocity? (0.577c) <Video>

10. A particle going 0.670c has a dilated mass of 147 MeV. What is its rest mass in MeV? (109 MeV) <Video>

11. An object going 0.850 c has a kinetic energy of 6.20x10¹⁴ J. What is its rest mass in kg? (0.00767 kg) <Video>

12. A particle has a rest mass of 410. MeV. How fast is it going if it has a kinetic energy of 110. MeV? (0.615c) <Video>

13. A 0.0850 kg ball has a 8.60x10¹⁵ J of kinetic energy. How fast is it going? (0.882c) <Video>

14. Through what potential (in volts) do you accelerated a proton so that it is going 0.850 c? (843x10⁶ V, 8.43x10⁸ V) <Video>

15. What is the kinetic energy of a 0.170 kg baseball going 0.820c? (answer in Joules) (1.14x10¹⁶ J) <Video>

16. What is the velocity of a 1.20 GeV (1GeV = 1000 MeV) Proton? (0.899c) <Video>

17. Problems from chapter 26: 24(9x10² kg), 26(0.866c), 28(2.23×10^-9 J, 6.46 × 10^-18 kg m/s), 29(1.51×10^-10 J, 8.70×10^-19 kg m/s), 31(0.437c), 32(0.941c), 33(0.30c), 37(5.5×10¹⁹ J), 35(0.866c, 0.745c)

Schwarzschild Radius: (R_s = radius of the event horizon - the point from which the V_esc > c)

18. What is the Schwarzschild radius of a 0.145 kg baseball? (2.15x10^-28 m) <Video>

19. What is the mass of a black hole with a radius of 1.00 m? (6.75x10²⁶ kg) <Video>

20. What is the Schwarzschild radius of a black hole with a mass of 21 million suns? (Msun = 1.99x10³⁰ kg) (6.19x10¹⁰ m) <Video>

21. What is the mass of a black hole that has an event horizon with a radius of 1.50x10¹¹ m? (1.01x10³⁸ kg) <Video>

Frequency Shift: (Low clocks run slowly)

22. A very strong concertmaster is playing 440.00 Hz at the top of an 4.50 m tall tower on a neutron star where the “g” is 1.816 x 10¹⁴ N/kg. We are at the bottom also playing 440.00 Hz. What is the beat frequency we hear? Do we hear the player on the top of the tower as sharp (higher frequency) or flat? What frequency do we observe? (4.00 Hz, Sharp, 444 Hz) <Video>

23. If we are living on a neutron star, and we tune the local station “Neutrock 91.7 (MHz) in at 90.2 on our FM Dial. We know that we are at a different elevation by 35.6 m. What is the “g” here? Are we higher or lower than the broadcast antenna of “Neutrock” (4.13x10¹³ N/kg, Higher) <Video>

24. A 417 nm spectral line is shifted to 423 nm through a distance of 1 A.U. What is the change in frequency? What is the “g” in the vicinity of source? (1.02 x 10¹³ Hz, 8520 N/kg) <Video>

Gravitational Time Dilation: (Low clocks run slowly) (Low = closer to the event horizon)

25. A black hole has a Schwarzschild radius of 39 km. What time does it take a clock 44 km from the event horizon to register 6.0 hours as we observe it from a distance? (8.2 hours) <Video>

26. A clock takes 173 minutes to register 120. minutes as we see it from a distance. It is 78.0 km from a black hole. What is the Schwarzschild radius of this black hole? What is its mass? (40.5 km, 2.73x10³¹ kg) <Video>

27. A star orbiting 89 km from a black hole has a 656 nm line spectral that has gravitationally red-shifted to 712 nm. What is the mass of the black hole? (Use v = fλ and f = 1/T to find the periods of 656 and 661 nm) (9.1x10³⁰ kg) <Video>

The Instruction for this unit: