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The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

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Linear Momentum and Collisions

Rotational Motion and Angular Momentum

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Fluid Statics

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Fluid Dynamics and Its Biological and Medical Applications

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Thermodynamics

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Oscillatory Motion and Waves

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Physics of Hearing

- Determine the appropriate number of significant figures in both addition and subtraction, as well as multiplication and division calculations.
- Calculate the percent uncertainty of a measurement.

The

The measurements in the paper example are both accurate and precise, but in some cases, measurements are accurate but not precise, or they are precise but not accurate. Let us consider an example of a GPS system that is attempting to locate the position of a restaurant in a city. Think of the restaurant location as existing at the center of a bull’s-eye target, and think of each GPS attempt to locate the restaurant as a black dot. In Figure 3, you can see that the GPS measurements are spread out far apart from each other, but they are all relatively close to the actual location of the restaurant at the center of the target. This indicates a low precision, high accuracy measuring system. However, in Figure 4, the GPS measurements are concentrated quite closely to one another, but they are far away from the target location. This indicates a high precision, low accuracy measuring system.

The factors contributing to uncertainty in a measurement include:

- Limitations of the measuring device,
- The skill of the person making the measurement,
- Irregularities in the object being measured,
- Any other factors that affect the outcome (highly dependent on the situation).

One method of expressing uncertainty is as a percent of the measured value. If a measurement *A* is expressed with uncertainty, δ*A*, the percent uncertainty (%unc) is defined to be

$\%\text{ unc}=\frac{\delta{A}}{A}\times100\%$

A grocery store sells a 5-pound bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements:

- Week 1 weight: 4.8 lb
- Week 2 weight: 5.3 lb
- Week 3 weight: 4.9 lb
- Week 4 weight: 5.4 lb

You determine that the weight of the 5-pound bag has an uncertainty of ±0.4 lb. What is the percent uncertainty of the bag’s weight?

First, observe that the expected value of the bag’s weight, *A*, is 5 lb. The uncertainty in this value, δ*A*, is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight:

$\%\text{ unc}=\frac{\delta{A}}{A}\times100\%$

Plug the known values into the equation:

$\%\text{ unc}=\frac{0.4\text{ lb}}{5\text{ lb}}\times100\%=8\%$

We can conclude that the weight of the apple bag is 5 lb ± 8%. Consider how this percent uncertainty would change if the bag of apples were half as heavy, but the uncertainty in the weight remained the same. Hint for future calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by 100%. If you do not do this, you will have a decimal quantity, not a percent value.

A high school track coach has just purchased a new stopwatch. The stopwatch manual states that the stopwatch has an uncertainty of ±0.05 s. Runners on the track coach’s team regularly clock 100-m sprints of 11.49 s to 15.01 s. At the school’s last track meet, the first-place sprinter came in at 12.04 s and the second-place sprinter came in at 12.07 s. Will the coach’s new stopwatch be helpful in timing the sprint team? Why or why not?

No, the uncertainty in the stopwatch is too great to effectively differentiate between the sprint times.

When we express measured values, we can only list as many digits as we initially measured with our measuring tool. For example, if you use a standard ruler to measure the length of a stick, you may measure it to be 36.7 cm. You could not express this value as 36.71 cm because your measuring tool was not precise enough to measure a hundredth of a centimeter. It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices that the stick length seems to be somewhere in between 36.6 cm and 36.7 cm, and he or she must estimate the value of the last digit. Using the method of significant figures, the rule is that

Determine the number of significant figures in the following measurements:

- 0.0009
- 15,450.0
- 6 × 10
^{3} - 87.990
- 30.42

(a) 1; the zeros in this number are placekeepers that indicate the decimal point

(b) 6; here, the zeros indicate that a measurement was made to the 0.1 decimal point, so the zeros are significant

(c) 1; the value 10^{3} signifies the decimal place, not the number of measured values

(d) 5; the final zero indicates that a measurement was made to the 0.001 decimal point, so it is significant

(e) 4; any zeros located in between significant figures in a number are also significant

*A* = π*r*^{2} = (3.1415927...) × (1.2 m)^{2} = 4.5238934 m^{2}

7.56 kg − 6.052 kg + 13.7 kg = 15.208 kg = 15.2kg

Next, we identify the least precise measurement: 13.7 kg. This measurement is expressed to the 0.1 decimal place, so our final answer must also be expressed to the 0.1 decimal place. Thus, the answer is rounded to the tenths place, giving us 15.2 kg.
Perform the following calculations and express your answer using the correct number of significant digits.

(a) A woman has two bags weighing 13.5 pounds and one bag with a weight of 10.2 pounds. 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/s^{2}, what is the force on the wagon? (The unit of force is called the newton, and it is expressed with the symbol N.)

(a) A woman has two bags weighing 13.5 pounds and one bag with a weight of 10.2 pounds. What is the total weight of the bags?

(b) The force

(a) 37.2 pounds; Because the number of bags is an exact value, it is not considered in the significant figures.

(b) 1.4 N; Because the value 55 kg has only two significant figures, the final value must also contain two significant figures.

(b) 1.4 N; Because the value 55 kg has only two significant figures, the final value must also contain two significant figures.

Explore size estimation in one, two, and three dimensions! Multiple levels of difficulty allow for progressive skill improvement.

- Accuracy of a measured value refers to how close a measurement is to the correct value. The uncertainty in a measurement is an estimate of the amount by which the measurement result may differ from this value.
- Precision of measured values refers to how close the agreement is between repeated measurements.
- The precision of a
is related to the size of its measurement increments. The smaller the measurement increment, the more precise the tool.*measuring tool* - Significant figures express the precision of a measuring tool.
- When multiplying or dividing measured values, the final answer can contain only as many significant figures as the least precise value.
- When adding or subtracting measured values, the final answer cannot contain more decimal places than the least precise value.

1. What is the relationship between the accuracy and uncertainty of a measurement?

2. 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.

**Express your answers to problems in this section to the correct number of significant figures and proper units.**

1. Suppose that your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)?

2. A good-quality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty?

3. (a) A car speedometer has a 5.0% uncertainty. What is the range of possible speeds when it reads 90 km/h? Convert this range to miles per hour. (1 km = 0.6214 m)

4. An infant’s pulse rate is measured to be 130 ± 5 beats/min. What is the percent uncertainty in this measurement?

5. (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?

6. A can contains 375 mL of soda. How much is left after 308 mL is removed?

7. 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).

8. (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?

9. (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?

10. (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?

11. A person measures his or her heart rate by counting the number of beats in 30s. If 40± 1 beats are counted in 30 ± 0.5 s, what is the heart rate and its uncertainty in beats per minute?

12. What is the area of a circle 3.102 in diameter?

13. If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22-mi marathon?

14. 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 1s 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?

15. The sides of a small rectangular box are measured to be 180 ± 0.01 cm long, 2.05 ± 0.02 cm, and 3.1 ± 0.1 cm long. Calculate its volume and uncertainty in cubic centimeters.

16. When non-metric units were used in the United Kingdom, a unit of mass called the * pound-mass* (lbm) was employed, where 11bm = 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?

17. 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.

18. A car engine moves a piston with a circular cross section of 7.500 ± 0.002 cm diameter in 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.

- accuracy:
- the degree to which a measured value agrees with correct value for that measurement

- method of adding percents:
- the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation

- percent uncertainty:
- the ratio of the uncertainty of a measurement to the measured value, expressed as a percentage

- precision:
- the degree to which repeated measurements agree with each other

- significant figures:
- express the precision of a measuring tool used to measure a value

- uncertainty:
- a quantitative measure of how much your measured values deviate from a standard or expected value

3. (a) 85.5 to 94.5 km/h (b)53.1 to 58.7 mi/h

5. (a) 7.6 × 10

7. (a) 3 (b) 3 (c) 3

9. (a) 2.2% (b) 59 to 61 km/h

11. 80 ± 3 beats/min

13. 2.6 h

15. 11 ± 1 cm

17. 12.06 ± 0.04 m