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Chapter 6: Q 4DQ (page 1519)
Under what conditions is average velocity equal to instantaneous velocity?
The average velocity is equal to instantaneous velocity when acceleration is zero. And also when a particle moves with constant velocity in a straight line.
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A particle is in the ground level of a box that extends from x= 0 to x= L. (a) What is the probability of finding the particle in the region between 0 and L/4? Calculate this by integrating , where c is normalized, from x= 0 to x= L/4. (b) What is the probability of finding the particle in the region x= L/4 to x= L/2 ? (c) How do the results of parts (a) and (b) compare? Explain. (d) Add the probabilities calculated in parts (a) and (b). (e) Are your results in parts (a), (b), and (d) consistent with Fig. 40.12b? Explain.
A sample of hydrogen atoms is irradiated with light with wavelength 85.5 nm, and electrons are observed leaving the gas. (a) If each hydrogen atom were initially in its ground level, what would be the maximum kinetic energy in electron volts of these photoelectrons? (b) A few electrons are detected with energies as much as 10.2 eV greater than the maximum kinetic energy calculated in part (a). How can this be?
Your company develops radioactive isotopes for medical applications. In your work there, you measure the activity of a radioactive sample. Your results are given in the table.
Time (h) Decays ,s
0 20,000
0.5 14,800
1.0 11,000
1.5 8130
2.0 6020
2.5 4460
3.0 3300
4.0 1810
5.0 1000
6.0 550
7.0 300
(a) Find the half-life of the sample.
(b) How many radioactive nuclei were present in the sample att= 0?
(c) How many were present after 7.0 h?
You are standing on a train platform watching a high-speed train pass by. A light inside one of the train cars is turned on and then a little later it is turned off. (a) Who can measure the proper time interval for the duration of the light: you or a passenger on the train? (b) Who can measure the proper length of the train car: you or a passenger on the train? (c) Who can measure the proper length of a sign attached to a post on the train platform: you or a passenger on the train? In each case explain your answer.
Many of the stars in the sky are actually binary stars, in which two stars orbit about their common centre of mass. If the orbital speeds of the stars are high enough, the motion of the stars can be detected by the Doppler shifts of the light they emit. Stars for which this is the case are called spectroscopic binary stars. Figure p=37.68 shows the simplest case of a spectroscopic binary star: two identical stars, each with mass m, orbiting their centre of mass in a circle of radius R. The plane of the stars鈥� orbits is edge-on to the line of sight of an observer on the earth. (a) The light produced by heated hydrogen gas in a laboratory on the earth has a frequency of . In the light received from the stars by a telescope on the earth, hydrogen light is observed to vary in frequency between and . Determine whether the binary star system as a whole is moving toward or away from the earth, the speed of this motion, and the orbital speeds of the stars. (Hint: The speeds involved are much less than c, so you may use the approximate result given in Section 37.6.) (b) The light from each star in the binary system varies from its maximum frequency to its minimum frequency and back again in 11 days. Determine the orbital radius R and the massm of each star. Give your answer form in kilograms and as a multiple of the mass of the sun, . Compare the value of to the distance from the earth to the sun, . (This technique is actually used in astronomy to determine the masses of stars. In practice, the problem is more complicated because the two stars in a binary system are usually not identical, the orbits are usually not circular, and the plane of the orbits is usually tilted with respect to the line of sight from the earth.)
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