Models in Relativity and Quantum Mechanics: An Explanation

Problem 1CQ Models are particularly useful in relativity and quantum mechanics, where conditions are outside those normally encountered by humans. What is a model?

Problem 1PE The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this?

Problem 2CQ How does a model differ from a theory?

Problem 2PE A car is traveling at a speed of 33 m/s . (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/h speed limit?

Problem 3CQ If two different theories describe experimental observations equally well, can one be said to be more valid than the other (assuming both use accepted rules of logic)?

Problem 3PE Show that 1.0 m/s = 3.6 km/h . Hint: Show the explicit steps involved in converting 1.0 m/s = 3.6 km/h.

Problem 4CQ What determines the validity of a theory?

Problem 4PE American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.)

Problem 5CQ Certain criteria must be satisfied if a measurement or observation is to be believed. Will the criteria necessarily be as strict for an expected result as for an unexpected result?

Problem 6CQ Can the validity of a model be limited, or must it be universally valid? How does this compare to the required validity of a theory or a law?

Problem 5PE Soccer fields vary in size. A large soccer field is 115 m long and 85 m wide. What are its dimensions in feet and inches? (Assume that 1 meter equals 3.281 feet.)

Problem 6PE What is the height in meters of a person who is 6 ft 1.0 in. tall? (Assume that 1 meter equals 39.37 in.)

Problem 7CQ Classical physics is a good approximation to modern physics under certain circumstances. What are they?

Problem 8CQ When is it necessary to use relativistic quantum mechanics?

Problem 8PE The speed of sound is measured to be 342 m/s on a certain day. What is this in km/h?

Problem 9CQ Can classical physics be used to accurately describe a satellite moving at a speed of 7500 m/s? Explain why or why not.

Problem 9PE Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year. (a) What distance does it move in 1 s at this speed? (b) What is its speed in kilometers per million years?

Problem 10CQ Identify some advantages of metric units.

Problem 11CQ What is the relationship between the accuracy and uncertainty of a measurement?

Problem 11PE Express your answers to problems in this section to the correct number of significant figures and proper units. Suppose that your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)?

Problem 12PE Express your answers to problems in this section to the correct number of significant figures and proper units. A good-quality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty?

Problem 13PE Express your answers to problems in this section to the correct number of significant figures and proper units. (a) A car speedometer has a 5.0% uncertainty. What is the range of possible speeds when it reads 90 km/h ? (b) Convert this range to miles per hour. (1 km = 0.6214 mi)

Problem 15PE Express your answers to problems in this section to the correct number of significant figures and proper units. a) Suppose that a person has an average heart rate of 72.0 beats/min. How many beats does he or she have in 2.0 y? (b) In 2.00 y? (c) In 2.000 y?

Problem 16PE Express your answers to problems in this section to the correct number of significant figures and proper units. A can contains 375 mL of soda. How much is left after 308 mL is removed?

Problem 14PE Express your answers to problems in this section to the correct number of significant figures and proper units. An infant’s pulse rate is measured to be 130 ± 5 beats/ min. What is the percent uncertainty in this measurement?

Problem 17PE Express your answers to problems in this section to the correct number of significant figures and proper units. State how many significant figures are proper in the results of the following calculations: (a)(106.7)(98.2) / (46.210)(1.01) (b) (18.7)2 (c) (1.60×10?19)(3712) .

Problem 18PE Express your answers to problems in this section to the correct number of significant figures and proper units. (a) How many significant figures are in the numbers 99 and 100? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers, significant figures or percent uncertainties?

Problem 19PE Express your answers to problems in this section to the correct number of significant figures and proper units. (a) If your speedometer has an uncertainty of 2.0 km/h at a speed of 90 km/h , what is the percent uncertainty? (b) If it has the same percent uncertainty when it reads 60 km/h , what is the range of speeds you could be going?

Problem 20PE Express your answers to problems in this section to the correct number of significant figures and proper units. (a) A person’s blood pressure is measured to be 120 ± 2 mm Hg . What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg ?

Problem 21PE Express your answers to problems in this section to the correct number of significant figures and proper units. A person measures his or her heart rate by counting the number of beats in 30 s. If 40 ± 1 beats are counted in 30.0 ± 0.5 s , what is the heart rate and its uncertainty in beats per minute?

Problem 23PE Express your answers to problems in this section to the correct number of significant figures and proper units. If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22-mi marathon?

Problem 22PE Express your answers to problems in this section to the correct number of significant figures and proper units. What is the area of a circle 3.102 cm in diameter?

Problem 24PE Express your answers to problems in this section to the correct number of significant figures and proper units. A marathon runner completes a 42.188-km course in 2 h , 30 min, and 12 s . There is an uncertainty of 25 m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?

Problem 25PE Express your answers to problems in this section to the correct number of significant figures and proper units. The sides of a small rectangular box are measured to be 1.80 ± 0.01 cm , 2.05 ± 0.02 cm, and 3.1 ± 0.1 cm long. Calculate its volume and uncertainty in cubic centimeters.

Express your answers to problems in this section to the correct number of significant figures and proper units. When non-metric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was employed, where 1 lbm = 0.4539 kg . (a) If there is an uncertainty of 0.0001 kg in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?

Problem 27PE Express your answers to problems in this section to the correct number of significant figures and proper units. The length and width of a rectangular room are measured to be 3.955 ± 0.005 m and 3.050 ± 0.005 m . Calculate the area of the room and its uncertainty in square meters.

Problem 28PE Express your answers to problems in this section to the correct number of significant figures and proper units. A car engine moves a piston with a circular cross section of 7.500 ± 0.002 cm diameter a distance of 3.250 ± 0.001 cm to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.

Problem 29PE How many heartbeats are there in a lifetime?

Problem 30PE A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD?

Problem 31PE How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of 10?22 s .)

Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an atom in the bacterium is ten times the mass of a hydrogen atom. (Hint: The mass of a hydrogen atom is on the order of \(10^{-27}\mathrm{\ kg}\) and the mass of a bacterium is on the order of \(10^{-15}\mathrm{\ kg}\). Equation Transcription: Text Transcription: 10^-27 kg 10^-15 kg

Problem 33PE Approximately how many atoms thick is a cell membrane, assuming all atoms there average about twice the size of a hydrogen atom?

Problem 34PE (a) What fraction of Earth’s diameter is the greatest ocean depth? (b) The greatest mountain height?

Problem 35PE (a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is ten times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human?

Problem 36PE Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?

Problem 12CQ Prescriptions for vision correction are given in units called diopters (D). Determine the meaning of that unit. Obtain information (perhaps by calling an optometrist or performing an internet search) on the minimum uncertainty with which corrections in diopters are determined and the accuracy with which corrective lenses can be produced. Discuss the sources of uncertainties in both the prescription and accuracy in the manufacture of lenses.