Chapter 1: Introduction
Problems & Exercises
1.1 What is Biomechanics?
- Roughly how many heartbeats are there in a lifetime?
- A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD?
- (a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is 10 times the mass of a bacterium.
(b) Making the same assumption, how many cells are there in a human? - Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?
1.2 Physical Quantities and Units
- The following times are given using metric prefixes on the base SI unit of time: the second. Rewrite them in scientific notation without the prefix. For example, 47 Ts would be rewritten as 4.7×1013s.
(a) 980 Ps
(b) 980 fs
(c) 17 ns
(d) 577μs - The following times are given in seconds. Use metric prefixes to rewrite them so the numerical value is greater than one but less than 1000. For example, 7.9×10−2s could be written as either 7.9 cs or 79 ms.
(a) 9.57×105s
(b) 0.045 s
(c) 5.5×10−7s
(d) 3.16×107s - The following lengths are given using metric prefixes on the base SI unit of length: the meter. Rewrite them in scientific notation without the prefix. For example, 4.2 Pm would be rewritten as 4.2×1015m.
(a) 89 Tm
(b) 89 pm
(c) 711 mm
(d) 0.45μm - The following lengths are given in meters. Use metric prefixes to rewrite them so the numerical value is bigger than one but less than 1000. For example, 7.9×10−2m could be written either as 7.9 cm or 79 mm.
(a) 7.59×107m
(b) 0.0074 m
(c) 8.8×10−11m
(d) 1.63×1013m - The following masses are written using metric prefixes on the gram. Rewrite them in scientific notation in terms of the SI base unit of mass: the kilogram. For example, 40 Mg would be written as 4×104kg.
(a) 23 mg
(b) 320 Tg
(c) 42 ng
(d) 7 g
(e) 9 Pg. - The following masses are given in kilograms. Use metric prefixes on the gram to rewrite them so the numerical value is bigger than one but less than 1000. For example, 7×10−4kg could be written as 70 cg or 700 mg.
(a) 3.8×10−5kg
(b) 2.3×1017kg
(c) 2.4×10−11kg
(d) 8×1015kg
(e) 4.2×10−3kg
1.3 Accuracy and Precision (and Significant Figures)
- Consider the equation 4000/400 = 10.0. Assuming the number of significant figures in the answer is correct, what can you say about the number of significant figures in 4000 and 400?
- Suppose your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)?
- A good-quality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty?
- An infant’s pulse rate is measured to be 130 ± 5 beats/min. What is the percent uncertainty in this measurement?
- (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 years?
(b) In 2.00 years?
(c) In 2.000 years? - 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) - (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? - A person’s blood pressure is measured to be 120±2mm Hg.
(a) What is its percent uncertainty?
(b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg? - 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?
- Determine the number of significant figures in the following measurements:
(a) 0.0009
(b) 15,450.0
(c) 6×103
(d) 87.990
(e) 30.42 - Perform the following calculations and express your answer using the correct number of significant digits.
(a) A woman has two bags weighing 13.5 lb and one bag with a weight of 10.2 lb. What is the total weight of the bags?
(b) The force F on an object is equal to its mass m multiplied by its acceleration a. If a wagon with mass 55 kg accelerates at a rate of 0.0255 m/s2, what is the force on the wagon? (The unit of force is called the newton and it is expressed with the symbol N.)
1.4 Conversions
- The volume of Earth is on the order of 1021 m3.
(a) What is this in cubic kilometers (km3)?
(b) What is it in cubic miles (mi3)?
(c) What is it in cubic centimeters (cm3)? - 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? - 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? - In SI units, speeds are measured in meters per second (m/s). But, depending on where you live, you’re probably more comfortable of thinking of speeds in terms of either kilometers per hour (km/h) or miles per hour (mi/h). In this problem, you will see that 1 m/s is roughly 4 km/h or 2 mi/h, which is handy to use when developing your physical intuition. More precisely, show that
(a) 1.0m/s=3.6km/h and
(b) 1.0m/s=2.2mi/h - American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 m = 3.281 ft.)
- Soccer fields vary in size. A large soccer field is 115 m long and 85.0 m wide. What is its area in square feet? (Assume that 1 m = 3.281 ft.)
- What is the height in meters of a person who is 6 ft 1.0 in. tall?
- Mount Everest, at 29,028 ft, is the tallest mountain on Earth. What is its height in kilometers? (Assume that 1 m = 3.281 ft.)
- The speed of sound is measured to be 342 m/s on a certain day. What is this measurement in kilometers per hour?
- Tectonic plates are large segments of Earth’s crust that move slowly. Suppose one such plate has an average speed of 4.0 cm/yr.
(a) What distance does it move in 1.0 s at this speed?
(b) What is its speed in kilometers per million years? - The average distance between Earth and the Sun is 1.5×1011m.
(a) Calculate the average speed of Earth in its orbit (assumed to be circular) in meters per second.
(b) What is this speed in miles per hour? - The density of nuclear matter is about 1018 kg/m3. Given that 1 mL is equal in volume to cm3, what is the density of nuclear matter in megagrams per microliter (that is, Mg/μL)?
- The density of aluminum is 2.7 g/cm3. What is the density in kilograms per cubic meter?
- A commonly used unit of mass in the English system is the pound-mass, abbreviated lbm, where 1 lbm = 0.454 kg. What is the density of water in pound-mass per cubic foot?
- A furlong is 220 yd. A fortnight is 2 weeks. Convert a speed of one furlong per fortnight to millimeters per second.
- It takes 2π radians (rad) to get around a circle, which is the same as 360°. How many radians are in 1°?
- Light travels a distance of about 3×108m/s. A light-minute is the distance light travels in 1 min. If the Sun is 1.5×1011m from Earth, how far away is it in light-minutes?
- A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.
Additional Problems
- 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 percent uncertainty in the elapsed time.
(c) What is the average speed in meters per second?
(d) What is the uncertainty in the average speed?