General Physics 1 - Accuracy, Precision, and Vector Addition

Accuracy and Precision

Objective

• Differentiate accuracy from precision.

• Differentiate random errors from systematic errors.

• Estimate errors from multiple measurements of a physical quantity using variance.

Review Questions

• Question 1: Which tool for measuring volume provides the most accurate measurement? Options: Beaker, Buret, Graduated cylinder, Electronic balance.

• Question 2: Three different people weight a standard mass of $2.00g$ on the same balance. Each person obtains a reading of exactly $7.32g$ for the mass of the standard. What does this imply about the balance? Options: Neither accurate nor precise, accurate but not precise, both accurate and precise, precise but not accurate.

• Question 3: Which of the following factors contribute to errors in measurement? Options: Limitation of the measuring device, the skill of the person making the measurement, the location where the measurement is done, irregularities in the object being measured.

• Question 4: If you do not keep your line of sight directly over length measurement, how will your measurement most likely be affected? Options: Your measurement will be less precise, Your measurement will be less accurate, Your measurement will have fewer significant figures, Your measurement will suffer from instrument error.

• Question 5: Your bathroom scale weighs you at $165 lb$, but you know that your true measurement is $172 lb$. What is the percentage error of your measurement? Options: $0.096\%$, $0.96\%$, $9.6\%$, $96\%$.

Significant Figures

• $572420$ has 6 significant figures.

• $0.0000350$ has 3 significant figures.

• $2370.0$ has 5 significant figures.

• $9.010 \times 10^{-9}$ has 4 significant figures.

• $101.0100$ has 7 significant figures.

Accuracy vs. Precision

• Illustrations depicting accuracy and precision using target diagrams: not accurate nor precise, precise but not accurate, accurate but not precise, and accurate and precise.

Accuracy

• Accuracy describes how close a measured value is to the true value of the quantity being measured.

• Problems with accuracy are due to an error.

Relative Error

• Accuracy is expressed using relative error.

• Example: The measured value of apples is $5.5 kg$, and the expected value is $8 kg$. What is the relative error?

Precision

• Precision: the degree of exactness with which a measurement is made and stated.

• An example is how close a second arrow is to the first.

Relative Uncertainty

• Precision is expressed as a relative or fractional uncertainty.

• Example: The mass of the apples is $5.5 kg$, and the measurement is $50\%$ uncertain. What is the relative or fractional uncertainty obtained?

Activity: Estimating Errors

• Problem 1: A student found the density of aluminum to be $2.85 g/cm^3$, while the accepted value is $2.70 g/cm^3$. What was the student's relative error? Answer: $5.56 \%$.

• Problem 2: A student weighed a calibrated $200.0$-gram mass on a balance and found it read $196.5 g$. What was the student's relative error? Answer: $1.75 \%$.

• Problem 3: A grocery store sells $2.268 kg$ bags of banana. Four bags are purchased over a month with the following measurements: Week 1: $2.177 kg$, Week 2: $2.404 kg$, Week 3: $2.223 kg$, Week 4: $2.449 kg$. The weight of the bag has an uncertainty of $\pm 0.181 kg$. What is the percent uncertainty of the bag's weight? Answer: $7.98 \%$.

Making Connections

• Discussing errors in location-based applications like Google Maps and assessing whether the places are accurate and precise.

Accuracy and Precision Scenarios

• Question 1: Which of the following situations shows high accuracy and high precision? Scenarios involve a basketball player and making baskets.

• Question 2: Which of the following situations shows high accuracy and low precision? Scenarios involve a basketball player and making baskets.

• Question 3: A student measures the mass of one mole of carbon and finds it to be $12.22$ grams. If the accepted value is $12.11$ grams, what is the student's % of error?

Vectors and Vector Addition

• Question 1: Which of the following represents a scalar quantity?

  • A load weighs $5$ Newton.

  • An airplane flies easterly of $100 km$.

  • A car has reached its destination after $1$ hour.

  • A car moves $60$ kilometers per hour at $35^o$ East of North.

• Question 2: What term represents the magnitude of a velocity vector?

  • Acceleration

  • Momentum

  • Speed

  • Velocity

• Question 3: What is the direction of the resultant vector a + b?

  • $15^o$ above the x-axis

  • $15^o$ below the x-axis

  • $75^o$ above the x-axis

  • $75^o$ below the x-axis

• Question 4: Vector A has a magnitude of $30$ units. Vector B is perpendicular to vector A and has a magnitude of $40$ units. What is the magnitude of vector A + B be?

  • $10$ units

  • $50$ units

  • $70$ units

  • zero units

Coordinate System & Sign Convention

• EAST

• WEST

• NORTH

• SOUTH

Scalar vs. Vector

Scalar vs. Vector Quantities

Graphical Method for Vector Addition

1. Construct an accurate Cartesian plane.

2. Plot the first force using a given scale.

3. At the end of the first force, construct another accurate Cartesian plane parallel to the first.

4. Plot the second force using the new Cartesian plane, and so on.

5. Connect the tail of the first force to the head of the last force and label it “R” for resultant.

6. To determine the magnitude of R, measure the length of R using the given scale.

7. To determine the direction of R, measure the angle with respect to the Y-axis.

Vector Addition Cases

• Case I: Two or more forces acting on an object in the same direction.

  • Example: F1 = $10 N$, due E; F2 = $20 N$, due E. Result: R = $30 N$, due E.

• Case II: Two or more forces acting on an object in opposite directions.

  • Example: F1 = $40 N$, due E; F2 = $60 N$, due W. Result: R = $20 N$, due W.

• Case III: Two forces acting on an object perpendicular to each other.

  • Example: F1 = $30N$, due E; F2 = $40N$, due S. Result: R = $50N$, $S37^\circ E$

• Case IV: Two or more forces acting on an object at random directions.

  • Example: F1 = $40N$ due E, F2 = $40N$ due E, F3 = $40N$ NE. Result: R = $116N$ $N76^\circ E$

• Case IV: Two or more forces acting on an object at random directions.

  • Example: F1 = $50N$ $50^o$ E of S = $50N$, S$50^o$E; F2 = $50N$ $30^o$ W of N = $50N$, N$30^o$W. Result: R = $19N$, N$48^\circ E$

Analytical Method

• Case III: Use Pythagorean Theorem and tangent.

  • R = \{Fx^2 + Fy^2}

  • $\theta = Tan^{-1} (\frac{F<em>y}{F</em>x})$

  • Example: F1 = $30N$ due E; F2 = $40N$ due S. $R = \sqrt{30^2 + 40^2} = 50N$. $\theta = Tan^{-1}(\frac{40}{30}) = 36.87$ E of S.

• Case IV: Construct a Cartesian plane without a need of a scale, plot the given vectors in the Cartesian plane, and use component method or sine and cosine law.

• Example:

  • $R = \sqrt{50^2 + 50^2 - 2(50)(50) \cos 20} = 17.36 N$

  • $\frac{\sin \beta}{50} = \frac{\sin 20}{17.36}$

  • $\beta = \sin^{-1} (\frac{50 \sin 20}{17.36}) = 80.09$

  • $\theta = 180 - (80.09 + 50)$

  • $\varnothing = N49.91^\circ E$