Physics Lab Review

A mathematical model is a description of a system using mathematical concepts and language.

A mathematical model?

C & D

A. A negative correlation between range (R) and the height of table (h) B. No correlation between range (R) and the height of table (h) C. A causal relationship between range (R) and the height of the table (h) D. A causal relationship between range (R) and the height of the table (h) if the horizontal velocity (v) and acceleration due to the gravity is held in constant

Something which is taken for granted; something that is believed to be true without proof.

Assumption?

Between days 1 and 4, neither student can claim that the mealworm grew because the ranges overlap on both graphs.

Can either student claim that the mealworm grew between days 1 and 4?

The relationship between two sets of variables used to describe or predict information. It can be positive, negative or non-correlation.

Correlation?

What I observe

Dependent Variable

What I change

Independent Variable

e. The amount of atmospheric carbon dioxide is related to the year it was measured.

a.All investigations measuring the amount of atmospheric carbon dioxide would yield similar results. b.The amount of atmospheric carbon dioxide is related to the location it was measured. c.If the levels of atmospheric carbon dioxide were measured in another observatory at the same latitude the same results would be found. d.The amount of atmospheric carbon dioxide is increasing each year. e.The amount of atmospheric carbon dioxide is related to the year it was measured.

c.The amount of product created is related to the temperature of the reaction.

a.The enzyme used is related to the temperature of the reaction. b.The amount of product created increases with temperature of the reaction. c.The amount of product created is related to the temperature of the reaction. d.The temperature of the reaction is related to the amount of product created. e.Enzyme B creates more product at a reaction temperature of 45

I agree most with Alex's statement. Response Feedback:Correct! Although inaccurate measurements might impact the results of the experiment, there are too many assumptions such as this one to report. The convention we use in this lab course is not to report those assumptions that refer to a student's ability to take and record data accurately.

A group of students conducted an experiment to determine what impacts the rate mold grows on bread. They decided to test whether or not preservatives in the bread made a difference in addition to the temperature in which the bread was stored. They purchased 2 loaves of bread that contain preservatives and 2 loaves that did not. They placed 5 pieces of each type of bread in 6 different environments of various temperatures ranging from 40°F to 85°F. The students were careful that each environment was basically identical except for temperature. After collecting data over 3 weeks the students made the claim that bread containing preservatives has a slower rate of mold growth when compared to bread without preservatives. They also made the claim that storing bread at lower temperatures reduces the rate of mold growth. Consider the following statements made by each student as they discuss what assumptions need to be discussed in their lab reports: Sam: In our report we need to write that we assumed that the two loaves of preservative-free bread were from the same batch and therefore identical to one another in terms of ingredients and age. We also need to write that we assumed the two loaves of bread with preservatives were identical in terms of ingredients and age. Alex: I agree with Sam but we should take it one step further and include a statement that we assumed that all four loaves of bread were actually of the same age and freshness as well. Troy: I agree with both of you, but we also need to include a statement that we assumed that our method of measuring the area of mold growth on each piece of bread was the most accurate.

The equation of the best fit line was given by Excel as . The equation can be 𝑦=0.9185𝑥+121.28 rewritten using V for volume and m for mass: . Using dimensional analysis, 𝑚=0.9185𝑉+121.28the units for the two numerical terms can be determined. The equation can then be written as: 𝑚=(0.9185gmL)𝑉+121.28 g The coefficient of the variable V is the density of the fluid; i.e. the density of the fluid is 0.9185 g/mL. It indicates the mass in grams of a unit volume (mL). The cylinder when empty has a mass of 121.28 g. That is, when the volume of liquid is zero (cylinder is empty), the mass is just the y-intercept which is 121.28 g. Or, one can use the above equation and plug in 0 mL for V and solve the equation for mass (m).

A group of students continually added liquid to a graduated cylinder that was sitting on a balance. With each addition of the liquid, they took a reading of the combined mass of the cylinder and liquid. They plotted their data and did a curve fit (linear). The equation of the best fit line provided by Excel is shown on the graph.What is the density of the liquid? What is the mass of only the graduated cylinder?

If volume is constant then the relationship between mass and density is causal (i.e. a change in total mass causes a change in density). [Note: if volume is NOT kept constant but a graph of density vs mass shows an upward trend, then all that can be said is that density is positively correlated to mass.]

A student finds that an increase of the mass of an object while keeping its volume fixed, such as when putting increasing amounts of water into a disposable water bottle, increases the density of the bottle/fluid system. What type of relationship exists between mass and density for a fixed volume- correlational (positive, negative, or none) or causal?

Because the balance was properly calibrated, it is likely that the error here is random error. The fluctuations in measured mass values could be due to a variety of issues including parallax in reading the scale (i.e. not looking head on when taking the measurement readings), scale not being truly balanced when reading was taken, etc. The student could minimize the impact of random error by taking multiple measurements and finding the average.

A student measures the mass of a rock three times using the same properly calibrated analog (not digital) balance and gets slightly different values: 22.46g, 22.42g, and 22.44g. What type of error may have occurred here (systematic and/or random)? How would the student minimize the impact of this error during data analysis?

We can't know for sure whether or not the mass changed, but it seems reasonable to claim that the mass did not change, given that the ranges of uncertainty overlap.

A student measures the mass of a solution before and after a chemical reaction takes place. In both cases the students measures the mass to be 50.25 g on an electronic balance with an uncertainty of 0.05 g. The student realizes that the ranges of uncertainty for each measurement overlap exactly. Which claim can the student make?

a. 4.90 m/s b. 49 N c. 98 N∙m d. 40 kg∙m2 e. 20 kg

A student wants to determine the mass of a pulley. He attaches a 10 kg box to it and lets it fall causing the pulley to rotate. The pulley is a solid disk of radius 2 m. He measures the angular acceleration of the disk to be 2.45 rad/s2. The questions will guide you to the answer. (Understanding this process will be useful for lab this week.) a. Determine the tangential acceleration of the pulley. b. Determine the net force on the pulley. This is essentially the tension in the string. c. Determine the net torque on the pulley. d. Determine the experimental moment of inertia I. e. Determine the mass of the pulley (consider it a solid disk).

Because the student noticed that the tape measure was stretched after years of use, this implies the presence of systematic error. To reduce this error the student needs to determine how much each measurement may have been off by comparing the original tape measure to one that has not been stretched out over years of use (such as a tape measure made of metal). Once the student knows how much each measurement is off, the student can adjust the original measurements accordingly. NOTE: Use of a tape measure relies on a person reading the scale, which introduces random error. Random error can be minimized by taking multiple measurements and calculating the average.

After collecting length data a student notices that the tape measure used was stretched out after years of use. What type of error may have occurred here (systematic and/or random)? How would the student minimize the impact of this error during data analysis?

At first glance this article title implies a causal relationship between buying ice cream and the number of drowning deaths. But, one must consider when people eat ice cream and when they actually swim more often. Both happen in the summer! So both are expected to increase due to the season of the year rather than one causing the other.

Article title from unnamed media: "As ice cream sales increase, the rate of drowning deaths increase."Are there other factors that might provide an alternative explanation? What is the type of relationship?

At first glance this article title implies a causal relationship and that taking HRT a woman will be less likely to develop CHD. But, one must consider the many other factors that have been linked to CHD such as access to healthcare, socio-economic status, social and behavioral risk factors. In the end another study controlled for some of these factors and determined that women that take HRT were actually more likely to develop heart disease.

Article title from unnamed media:"Women that take hormone replacement therapy (HRT) are less likely to have coronary heart disease". Are there other factors that might provide an alternative explanation? What is the type of relationship?

Between days 4 and 10, both students can claim that the mealworm grew because the ranges do not overlap on either graph for these days.

Can either student claim that the mealworm grew between days 4 and 10?

Between days 4 and 7, student 1 can claim that the mealworm grew because the ranges do not overlap on his graph. However, student 2 cannot claim that the mealworm grew because the ±2σranges overlap for these two days.

Can either student claim that the mealworm grew between days 4 and 7?

also known as cause and effect, is when an observed event or action appears to have caused a second event or action. For example, I bought a brand new bed comforter and placed it in my washing machine to be cleaned. After cleaning the comforter, my washing machine stopped working. I may assume that the first action, washing the comforter, caused the second action, broken washing machine.

Causation?

1. Systematic errors in experimental observations usually come from the measuring instruments. 2. Random errors in experimental measurements are caused by unknown and unpredictable changes in the experiment. These errors are fluctuations in measurements inherent to the measuring device itself or to the chosen measurement technique.

Causes of measurement error

The thin spherical shell has a larger moment of inertia as given in Table 1 so it will require more force (torque) to start it rotating from rest. This should make sense as both objects have the same mass which means the shell has more mass further away from the axis of rotation making it harder to rotate. This also means that once rotating, the shell will be harder to stop rotating!

Consider a hollow and solid sphere of the same mass and radius. If both have an axis of rotation through their centers, which would take more force to start rotating from rest?

No correlation between nightlights and nearsightedness. One should question what causes nearsightedness! It is already known that there is a strong link between parental nearsightedness and child nearsightedness. Interestingly, a follow up study to this report in Nature showed nearsighted parents were more likely to leave the light on in a child's room, so this introduces doubt in the article finding and reduces the relationship to one of positive correlation.

Consider an article title from the May 13, 1999 edition of the journal Nature: "Children that sleep with the light on are likely to develop nearsightedness later in life". Are there other factors that might provide an alternative explanation? What is the type of relationship?

To answer this question, each student must first determine the range of uncertainty for their final values to decide if the ranges overlap with the true (accepted) value of 9.80 m/s2. Because statistically 95% of the measurements fall within two standard deviations, the students might choose to use ±2σ in computing their ranges of uncertainty. In this way the students are able to claim at the 95% confidence level that their ranges actually include the true value. Note: In this lab course we will use ±2σ when computing comparison ranges, as this yields a range of values for which there is a high level of confidence that the true value lies within it. However, remember that standard convention is to report measurements with just one standard deviation; that is, mean ±σ. Student A: - She calculates the ±2σ range for the measurement to be: (9.77 - 0.1, 9.77 + 0.1) m/s2, where the 0.1 is found from 2σ = 2(0.05). - She calculates the ±2σ range for the accepted value to be the exact value 9.80 m/s2(σ = 0.00). - The computed range goes from 9.67 to 9.87 m/s2 and includes the accepted value. Student B: - She calculates the ±2σ range for the measurement to be: (9.75 - 0.04, 9.75 + 0.04) m/s2, where the 0.04 is found from 2σ = 2(0.02). - She calculates the ±2σ range for the accepted value to be the exact value 9.80 m/s2 (σ = 0.00). - The computed range goes from 9.69 to 9.79 m/s2 and does not include the accepted value. => student A

Consider an experiment conducted by two students who measure the acceleration due to gravity. Student A finds a value for g to be 9.77 ± 0.05 m/s2, and student B finds a value for g to be 9.75 ± 0.02 m/s2. Which student, if any, is consistent with the accepted value of 9.80 m/s2 for the acceleration due to gravity in Cincinnati, where the experiment was conducted? Note that the "true value" of 9.80 m/s2 is a more precisely measured value that has been rounded to three significant figures, so no range of uncertainty is reported here for this value. The ±2σ range describes the region where approximately 95% of measurements should fall.

Answer:c. Although inaccurate measurements might impact the results of the experiment, this is not considered an appropriate assumption to report. Response Feedback:Correct! Although inaccurate measurements might impact the results of the experiment, there are too many assumptions such as this one to report. The convention we use in this lab course is not to report those assumptions that refer to a student's ability to take and record data accurately.

Consider the following assumption made by the students:"The amount of coffee grounds was measured accurately before adding them to the soil."is it appropriate?

This is an appropriate assumption because the health of the plant might impact leaf growth.Response Feedback:Correct! If the seedlings were not of similar health then the leaf growth might be affected by this factor. This is a reasonable inference based on past experiences even though the students may not be able to check the health of each plant used in the experiment.

Consider the following scenario: According to Sustainable Enterprises, coffee grounds can greatly benefit plants. They allow for a slow release of nitrogen and they can also increase nitrogen balance. Nitrogen helps plants use carbohydrates to gain energy. Nitrogen controls how plants take their form and how they function inside, and nitrogen helps plants make proteins that help them grow strong and healthy.Coffee grounds have been shown to increase the growth of plants because they have been said to release important nutrients used by the plants. According to Grow Joe, coffee grounds also release magnesium and zinc, micronutrients and amino acids. Without enough magnesium, plants may have brown/yellow older leaves. The coffee grounds can also feed earthworms; they loosen the soil; they retain water; and they release caffeine which repels slugs.Based on this information, students hypothesized that adding coffee grounds to the soil would affect the growth of Brassica rapa plants as measured by their leaf mass (size of leaves grown). They tested this by adding different amounts of coffee grounds to pots of soil before transplanting Brassica rapa seedlings into the pots. These amounts included 0, ¼, ½, ¾, and 1 cup of coffee grounds in each pot. Consider the following assumption made by the students:"The Brassica rapa seedlings transplanted into each pot were similar in health

Be careful! The claim is incorrect. The theoretical model indicates that the period is 𝑻=𝟐𝝅𝑳/𝒈actually determined by two variables, length (L) and acceleration due to gravity (g). This implies that the causal relationship is between T and two variables taken together (L and g). Therefore, the student would have been correct if he had said that the length and period have a causal relationship when g is held constant. When g is not held constant, such as when comparing the period of a pendulum on the moon versus one on the earth, the best the student can claim is that period and length are positively correlated.

Consider the student, who after determining the experimental mathematical model for the period of a pendulum and comparing it to the theoretical model of , makes the claim that because the 𝑇=2𝜋𝐿/𝑔length impacts the period of the pendulum but mass and angle of release (for small angles) do not, that length (L) and period (T) demonstrate a causal relationship. That is, a change in length causes a change in the period. Do you agree with this statement?

(b)

Consider two identical rods rotating around two different axes as shown in Figure 2. If each rod were spinning at the same rev/min, which would be easier to stop? ____________

What I keep the same

Control Variable

"The __________ (DV) is affected by (or related to) the ______________ (IV)." "The __________ (DV) is directly related to the ______________ (IV)." "The __________ (DV) is inversely related to the ______________ (IV)."

Hypothesis structure

1/4

If the radius of a thin cylindrical shell were doubled, by how much would the mass of the shell need to be changed to maintain its original moment of inertia ( ) if the axis of rotation passes through it center 𝐼of mass as shown in Table 1? The moment of inertia

This is false. The moment of inertia depends on more than just the mass of the object. It also depends on the position of the axis of rotation for the object, so it has as many moments of inertia as there are possible axes of rotation.

Is the statement below true or false:Every object has a single mass, so every object has a single moment of inertia.

→ Changing the mass riding on the cart on the air track does not change the amount of friction between the cart and the track. We could test this by observing how far a differently-massed cart goes on the track with a small push and no hanging mass. A different frictional force would change how far the cart could travel. → The air flow from the holes on the track merely lifts the cart off the track and does not exert any forward or backward forces on the cart. We could test for this by placing the cart on the track and noting if it remains at rest even after the track has been properly leveled. → If friction is present, it is the same everywhere on the track. → The track is exactly straight and not curved. → The string is massless. → The pulley is massless.

Make a list of assumptions made in the Force and Motion experiment conducted during Labs 04 and 05.

→ The pendulum string is massless. → The pendulum bob behaves like a point mass. → The geometrical center of the brass cylinder (bob) is the center of mass. → The system is frictionless. → The string doesn't stretch. → The arm of the pendulum support does not bend as the bob swings.

Make a list of assumptions made in the pendulum lab conducted at the beginning of the term during Labs 01, 02, and 03.

Taking into account the ranges of uncertainty for each measurement, the ranges for each day are found using 2σ where σ = 0.05 g in this example: Monday: The measurement of 150.5 ± 0.05 g suggests a range of (150.5 - 0.10 g, 150.5 + 0.10 g) Tuesday: The measurement of 150.4 ± 0.05 g suggests a range of (150.4 - 0.10 g, 150.4 + 0.10 g) Friday: The measurement of 150.1 ± 0.05 g suggests a range of (150.1 - 0.10 g, 150.1 + 0.10 g) The student's claim will therefore be that from Monday to Tuesday, due to the overlap of the two mass ranges, the claim cannot be made that the mass decreased. However, the range of possible masses on Friday does not overlap either of the other ranges indicating that the mass did decrease during the week. The student may use prior knowledge and infer this change in mass is likely due to evaporation (unless other experimental conditions indicate otherwise).

On Monday, a student measures the mass of an open beaker of water to be 150.5 g on an electronic balance with an associated uncertainly of 0.05 g. On Tuesday, the student uses the same balance and measures the mass to be 150.4 g, and on Friday it is measured to be 150.1 g. What conclusion can the student make in regards to whether mass was lost during the week due to evaporation? Ironically, scientists can make completely certain statements or claims once uncertainties are known. This scenario is an example of that!

A causal relationship exists between the distribution of mass of an object and its moment of inertia (I)

The differences between causal relationships and correlations were explored. The moment of inertia I, the best statement for single object of the mass m

C. The propeller can be modeled as a rod that rotates about its center. The engine exerts a torque on the propeller.

The engine in a small airplane is specified to have a torque of 60 N•m. This engine drives a 2.0 m long, 40 kg propeller. How can you model this in terms of moments of inertia and what exactly gets the propeller spinning? A. The propeller can be modeled as a rod that rotates about one end. The engine exerts an angular acceleration on the propeller. B. The airplane can be modeled as a rotating cylinder about one end. The engine exerts an angular acceleration on the propeller. C. The propeller can be modeled as a rod that rotates about its center. The engine exerts a torque on the propeller. D. The engine can be modeled as rotating drum. The engine exerts a torque on the propeller. E. The propeller can be modeled as a rotating cylinder. The engine exerts a forward force on the airplane.

The relationship between SAT score and student success can only be claimed to be correlated. No variables were manipulated here to establish a causal relationship. This also means that one should not interpret the results of this study to mean that if you have a high SAT score that you will be successful in biology. It could be that students with high SAT scores are also those who in general study more or attend class more often.

Two educational researchers find that higher student SAT scores result in higher student success in an introductory biology course. What type of relationship exists between SAT scores and student success- correlational (positive, negative, or none) or causal?

No, they are not correct. The students mixed up the variables for and here. They should have 𝑦𝑥instead determined the relationship.

Two students wish to determine the mathematical relationship between thermal resistance (R) and air velocity (Vair). Their data is plotted on the graph below. They decided that the data best matched a power curve (see options for common graphs later in this document) and chose that option in Excel. The resulting equation for the best fit curve, provided by Excel, is shown on the graph. The students rewrote this equation as: 𝑉air=400/𝑅^0.25 Is their equation correct?

Uncertainty as used here means the range of possible values within which the true value of the measurement lies.

Uncertainty?

The length falls between 128.8 and 128.9 cm, with it being closer to 128.9. A common convention is to estimate one digit further than what is known. Therefore, a reading of 128.88 cm or 128.89 cm is reasonable. The uncertainty is estimated as half the smallest division on the measuring device. The smallest division is 0.1 cm so the reported uncertainty is 0.05 cm. This measurement is reported as 128.88 ± 0.05 cm. Note that there are different conventions but this is the one we will use in lab.

What measurement and uncertainty should be reported for the length of a pendulum measured from one end of the string at 0.00 mm to the bottom of the bob at the other end, shown here? The smallest division on the ruler shown below is 1 mm.

Correct! Listing assumptions and their possible effects on the experiment, as well as suggestions to mitigate these effects, is good scientific practice.

When discussing assumptions in lab reports for this course, which of the following is the best course of action that you should take?

The rotational analog to force 𝜏= 𝐼𝛼 = 𝑟𝐹sin𝜃

torque τ?