Chapter 1. Measurement

I. Units of Mass, Time, and Length

1. Which of the following is the unit of length?

a) meter (m);

b) kilogram (kg);

c) second (s);

d) Ampere (A);

e) Kelvin (K).

2. Which of the following is the unit of mass?

a) meter (m);

b) kilogram (kg);

c) second (s);

d) Ampere (A);

e) Kelvin (K).

3. Which of the following is the unit of time?

a) meter (m);

b) kilogram (kg);

c) second (s);

d) Ampere (A);

e) Kelvin (K).

4. The meter is a unit used to measure:

a)length;

b)mass;

c)time;

d)electric current;

e)temperature.

5. The kilogram is a unit used to measure:

a)length;

b)mass;

c)time;

d)electric current;

e)temperature.

6. The second is a unit used to measure:

a)length;

b)mass;

c)time;

d)electric current;

e)temperature.

7. Which of the following units are used to measure length (choose three)?

a) meter (m);

b) second (s);

c) terameter (Tm);

d) millimeter (mm);

e) gram (g);

f) milligram (mg).

8. Which of the following units are used to measure time (choose two)?

a) nanosecond (ns);

b) gigasecond (Gs);

c) kilometer (km);

d) kilogram (kg);

e) Kelvin (K).

9. Which of the following units are used to measure mass (choose three)?

a) kilogram (kg);

b) second (s);

c) terameter (Tm);

d) millimeter (mm);

e) gram (g);

f) milligram (mg).

II. Scientific Notation

  1. In scientific notation, the number 12,300 would be written as:

a)123 x 102;

b)12.3 x 104;

c)1.23 x 105;

d)1.23 x 106;

e)1.23 x 107.

Hint 1/Comment:

Scientific notation is a mathematical shorthand. It is represented by a base (a number that has a single nonzero digit to the left of the decimal), which is multiplied by a power of 10.

  1. In scientific notation, the number 2,300,000 would be written as:

a)2.3 x 103;

b)230 x 104;

c)23 x 105;

d)2.3 x 106;

e)2.3 x 107.

Hint 1/Comment:

Scientific notation is a mathematical shorthand. It is represented by a base (a number that has a single nonzero digit to the left of the decimal), which is multiplied by a power of 10.

  1. In scientific notation, the number 37,000 would be written as:

a)37 x 103;

b)3.7 x 104;

c)0.37 x 105;

d)0.37 x 106;

e)3.7 x 107.

Hint 1/Comment:

Scientific notation is a mathematical shorthand. It is represented by a base (a number that has a single nonzero digit to the left of the decimal), which is multiplied by a power of 10.

  1. In scientific notation, the number 7,420 would be written as:

a)7.42 x 103;

b)0.742 x 104;

c)0.0742 x 105;

d)742 x 106;

e)7.42 x 107.

Hint 1/Comment:

Scientific notation is a mathematical shorthand. It is represented by a base (a number that has a single nonzero digit to the left of the decimal), which is multiplied by a power of 10.

  1. In scientific notation, the number 11,900,000 would be written as:

a)1.19 x 103;

b)1.19 x 104;

c)119 x 105;

d)11.9 x 106;

e)1.19 x 107.

Hint 1/Comment:

Scientific notation is a mathematical shorthand. It is represented by a base (a number that has a single nonzero digit to the left of the decimal), which is multiplied by a power of 10.

  1. In scientific notation, the number 0.0041 would be written as:

a)4.1 x 10-3;

b)41 x 10-4;

c)410 x 10-5;

d)4.1 x 10-6;

e)4.1 x 10-7.

Hint 1/Comment:

Scientific notation is a mathematical shorthand. It is represented by a base (a number that has a single nonzero digit to the left of the decimal), which is multiplied by a power of 10.

  1. In scientific notation, the number 0.0000057 would be written as:

a) 5.7 x 10-3;

b) 570 x 10-8;

c) 5.7 x 10-5;

d) 5.7 x 10-6;

e) 57 x 10-7.

Hint 1/Comment:

Scientific notation is a mathematical shorthand. It is represented by a base (a number that has a single nonzero digit to the left of the decimal), which is multiplied by a power of 10.

  1. Written in scientific notation, 1.8 x 102 represents the number:

a) 180;

b) 1,800;

c) 18,000;

d) 180,000;

e) 1,800,000.

  1. Written in scientific notation, 5.6 x 108 represents the number:

a)5,600;

b)56,000;

c) 560,000;

d) 560,000,000;

e) 5,600,000,000.

  1. Written in scientific notation, 2.37 x 102 represents the number:

a)23.7;

b)237;

c) 2,370;

d) 23,700;

e) 23,700,000.

III. Metric Prefixes

Common Prefixes

  1. The metric prefix centi means:

a)101;

b)10-1;

c)10-2;

d)10-3;

e)10-4.

2. The metric prefix kilo means:

a)10-1;

b)101;

c)102;

d)103;

e)104.

3. The metric prefix milli means:

a)10-2;

b)10-3;

c)10-4;

d)10-5;

e)10-6.

4. The metric prefix micro means:

a)10-2;

b)10-3;

c)10-4;

d)10-5;

e)10-6.

5. The metric prefix mega means:

a)102;

b)103;

c)104;

d)105;

e)106.

6. The metric prefix giga means:

a)106;

b)107;

c)108;

d)109;

e)1010.

Uncommon Prefixes

7. The metric prefix peta means:

a)102;

b)103;

c)104;

d)105;

e)none of these.

8. The metric prefix deka means:

a)103;

b)102;

c)101;

d)10-1;

e)10-2.

9. The metric prefix tera means:

a)1011;

b)1012;

c)1013;

d)1014;

e)1015.

10. The metric prefix hecto means:

a)102;

b)103;

c)104;

d)105;

e)106.

11. The metric prefix pico means:

a)10-12;

b)10-13;

c)10-14;

d)10-15;

e)10-16.

Applications of Prefixes

12. Which of the following prefixes are greater than 1000 (choose three)?

a) peta (P);

b) giga (G);

c) milli (m);

d) micro ();

e) centi (c);

f) mega (M).

13. Which of the following prefixes are less than 1 (choose two)?

a) hecto (h);

b) femto (f);

c) mega (m);

d) deci (d);

e) deka (da);

f) kilo (k).

14. Which of the following quantities give 1 megameter (Mm) if multiplied or divided by 103 (choose two)?

a) 1 gigameter (Gm);

b) 1 terameter (Tm);

c) 1 kilometer (km);

d) 1 nanometer (nm);

e) 1 decimeter (dm).

15. Which of the following prefixes become a larger (smaller) number when squared (cubed)? (Choose three).

a) deka (da);

b) kilo (k);

c) tera (T);

d) femto (f);

e) micro ();

f) deci (d).

IV. Analysis of Dimensions

  1. What are the dimensions of the left (right) side of the equation, where a is acceleration (measured in m/s2), x is position (measured in m), and v is velocity (measured in m/s)?
  2. Velocity (measured in m/s) as a function of time is expressed by the following function , where t is time (in seconds, s). Find the dimensions of p.

Hint 1/Comment:

Let us suppose the units of p to be [z]. Substitute this and the units of other quantities in the original equation.

Hint 2/Comment:

In terms of units, the original equation is written as follows:

.

Comment:

Solving this equation for [z], we obtain the units of p:

  1. Acceleration (measured in m/s2) is related to time and distance by the following formula: , where x is position (in m/s) and t is time (in seconds, s). Find the value of n that makes this equation dimensionally consistent.

Hint 1/Comment (Hint 1 does not have the last step):

Substituting the units of all quantities in the given equation, we obtain:

.

Note that  has no units.

In order for this equation to be true, the powers of respective units of measurement must be equal, so to find n, we need to solve 2 = n/4, which gives n = 8.

  1. The frequency (measured in 1/s) of oscillation of a mass on a spring is given by , where m is the mass (measured in kg). Find the dimensions that the constant k must have in order for this equation to be dimensionally accurate. (Give your answer in basic SI units.)

Hint 1/Comment:

Let us suppose the units of k to be [z]. Substitute this and the units of other quantities in the original equation.

Hint 2/Comment:

In terms of units, the original equation is written as follows:

Note that  is dimensionless (has no units).

Comment:

Solving this equation for [z], we obtain that the units of k are kg/s2.

  1. The velocity (measured in m/s) of a particle is described by the following formula: , where t is time (in seconds, s) and a is acceleration (measured in m/s2). What dimensions must k have in order for that equation to be dimensionally correct?

Hint 1/Comment:

Rewrite the given equation in terms of dimensions of the quantities using [k] to represent the dimensions of k.

Hint 2/Comment:

In terms of units the equation becomes:

Because solving for either [k] will give the same result, we can use either of the terms on the right side of the equation. Let us choose the second term and ignore the first one. Thus the equation becomes:

Solving this equation for [k] will give the final answer.

Comment:

[k] = s2.

  1. Position (measured in meters, m) is related to time t (measured in seconds, s), acceleration a (measured in m/s2), and instantaneous velocity v (measured in m/s) according to the following formula: (where is an exponential function). Find the dimensions of b so that the equation is dimensionally consistent.

Hint 1/ Comment:

Rewrite the given equation in terms of dimensions of the quantities using [b] to represent the dimensions of b.

Hint 2/Comment:

In terms of units the equation becomes:

Hint 3/Comment:

Because exponents are dimensionless, the units in the exponent must cancel.

Comment

In order to cancel the units of velocity (m/s), the units of b must be s/m.

  1. A time interval in an experiment can be calculated using the formula , where v is velocity (measured in m/s), a is acceleration (measured in m/s2), and t is measured in seconds, s. Find the dimensions of the constant z that would keep the equation dimensionally correct.

Hint 1

Rewrite the given equation in terms of the dimensions of the quantities. Replace z with [z] to indicate the respective dimensions of the quantity z.

Hint 2

In terms of units, the given equation becomes . Now we need to solve for [z].

Hint 3

Solving for [z], we get:


  1. What dimensions must the constant k have in order for the equation to be dimensionally correct? (Note that v is velocity measured in m/s, x is position measured in meters, m).

Hint 1/Comment:

Rewrite the equation in terms of dimensions of the quantities. Use [k] to represent the dimensions of k.

Hint 2/Comment:

In terms of units, the equation becomes:

In order to subtract the terms on the right side of the equation they must be dimensionally consistent. Note that the numerical constant can be ignored.

Comment:

V. Significant Figures

  1. How many significant figures does the number 0.004 (0.0040, 0.00400, 0.004000, 0.0040000) contain?
  2. How many significant figures does the number 0.06 (0.060, 0.0600, 0.06000, 0.060000) contain?
  3. How many significant figures does the number 40 (36, 36.0, 36.01, 36.005) contain?
  4. How many significant figures does the number 7 (7.0, 7.00, 7.004, 7.0040) contain?
  5. How many significant figures does the number 10 (14, 14.0, 14.00, 14.000) contain?
  6. How many significant figures does the number 2 x 104 (2.4 x 104, 2.40 x 104, 2.400 x 104, 2.4000 x 104) contain?
  7. How many significant figures does the number 500 (530, 534, 534.0, 534.00) contain?
  8. How many significant figures does the number 6 x 102 (6.2 x 102, 6.22 x 102, 6.220 x 102, 6.2200 x 102) contain?
  9. How many significant figures does the number 7 x 106 (7.4 x 106, 7.42 x 106, 7.421 x 106, 7.4210 x 106) contain?

VI. Unit Conversions

  1. Suppose that the average period of life of a human being is 67 years. Express this amount of time in seconds. (Suppose 1 year to be 365 days).
  2. The speed of light in vacuum is equal to. Express this speed in meters per kilosecond, m/ks (nanometers per second, nm/s).
  3. The acceleration a of a space ship is equal to 4.33g, where g is acceleration due to gravity and is equal to 9.81 m/s2. Express a in km/s2 (km/h2, m/h2).

Hint 1/Comment:

When converting between squared units, the conversion factor used must also be squared.

  1. A surface area of a side of the cube is equal to 10 m2. What is its volume expressed in liters, l (cm3)?

Hint 1/Comment:

The volume of a cube is equal to its side cubed: V = s3. Solve for the length of the side of the cube first.

Hint 2/Comment:

When converting between cubed units, the conversion factor must also be cubed.

  1. The diameter of our galaxy (distance from Earth to the nearest large galaxy, distance from Earth to the nearest star, distance from Earth to the Sun, radius of Earth) is 8 x 1020 (2 x 1022, 4 x 1016, 1.5 x 1011, 6.37 x 106) meters. How many hours, h, will it take a ray of light to travel that distance?
  2. The speed limit on highways is 65 mi/h. What is this speed limit written in ft/s (feet per second)?

a)75 ft/s;

b)85 ft/s;

c)95 ft/s;

d)105 ft/s;

e)115 ft/s.

  1. To convert km/h into m/s, you should:

a)multiply by 1000 and multiply by 3600;

b)divide by 1000 and multiply by 3600;

c)multiply by 1000 and divide by 3600;

d)divide by 1000 and divide by 3600.

  1. The speed of sound at room temperature is 340 m/s. What is the equivalent of this value in miles per hour (mi/h)?

a)167 mi/h;

b)324 mi/h;

c)423 mi/h;

d)584 mi/h;

e)942 mi/h.

  1. A common highway speed limit in a foreign country is 175,000 furlongs per fortnight. Given that one fortnight is 14 days and a furlong is 1/8 of a mile, what is this speed limit in mi/hr?

a)35 mi/h;

b)45 mi/h;

c)55 mi/h;

d)65 mi/h;

e)75 mi/h.

VII. Order of Magnitude Problems

  1. How many Physics textbooks would you have to stack to reach the Moon?

a)104;

b)107;

c)1010;

d)1013;

e)1014.

  1. An aquarium is 81.4 yd long, 61.0 yd wide and 11.2 yd high. Approximately how many cans of soda will it take to fill the aquarium with the drink?

a)1.5 x 103;

b)1.5 x 108;

c)1.5 x 1013;

d)1.5 x 1018;

e)1.5 x 1023.

Hint 1/Comment:

A soda can could be considered a cylinder. The volume of a cylinder is V = r2h, where r is the radius of the top (or the bottom) of the cylinder, and h is the height.

  1. How many compact disks would you have to stack on top of each other in order for the height of the stack to be equal to the height of a personal computer case?

a)3 x 101;

b)3 x 102;

c)3 x 103;

d)3 x 104;

e)3 x 105.

  1. How many Kleenex tissues would you have to place side by side in order to cover the entire surface of the Earth?

a)1013;

b)1016;

c)1019;

d)1022;

e)1025.

Hint 1/Comment:

Surface area of a sphere is A = 4r2.

  1. How much of the tire tread is worn down after a single revolution of the wheel? Suppose that the average life of a tire is 65,000 miles.

a)2 x 10-4 m;

b)2 x 10-6 m;

c)2 x 10-8 m;

d)2 x 10-10 m;

e)2 x 10-12 m.

Hint 1/Comment:

Calculate how many revolutions of the wheel there are in 65,000 miles.

Hint 2/Comment:

Circumference of a circle equals 2r.

  1. What is the closest estimate of the number of times that a human heart beats in a lifetime?

a)2 x 103;

b)2 x 105;

c)2 x 107;

d)2 x 109;

e)2 x 1011.

VIII. Review Problems

  1. The length of a rectangular garden of flowers is L=14.2 yd and its width is W=3.15 yd. What is its perimeter in inches (in)?
  2. Assuming that the Earth is a perfect sphere, find its volume in cm3 (in3, yd3, m3, ft3). (Radius of the Earth is ).

Hint 1/Comment:

Volume of a sphere equals 4/3*r3.

  1. If a radius of a circle is increased by 60%, what factor is its area increased by?

Hint 1/Comment:

The area of a circle equals r2.

Comment:

As radius increases, the area increases by that factor squared.

  1. Find the circumference (in km) of 1/5 of a circle with the radius of 15 m.

Hint 1/Comment:

The circumference of a circle is 2r.

  1. What is the volume (in liters, l) of a pancake that is 0.7 cm thick and has a radius of 12 cm?

Hint 1/Comment:

The pancake can be treated as a cylinder. Volume of a cylinder is r2h, where r is the radius of the top (or the bottom of the cylinder) and h is the height.

  1. Find the larger diagonal (in terameters, Tm) of a cube that has a side of 16.6 m.

Hint 1/Comment:

Hint 2/Comment:

AB is the larger diagonal we are asked to find. AC is the length of the side of the cube and CB is the diagonal of the base of the cube. Use Pythagorean Theorem to find AB.

  1. How many gigaseconds, Gs, are there in a day (year, century, millennium)?
  2. What is the volume of air, in cubic feet, of a room whose dimensions are

Hint 1/Comment:

When converting between cubic dimensions, the conversion factor must also be cubed.

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