Measuring Gravity

In this laboratory we will measure the acceleration due to gravity by studying the motion of a cart accelerating down an inclined plane.

Background
Suppose we start with a level track and then tip it, as shown in Figure 1 below. Let L be the distance between two fixed points on a ramp, selected to be as far apart as possible, on the track. Let h be the difference in the vertical height above the table of these two points.

Figure 1 – Schematic of a cart on an inclined plane. The magnitude of the acceleration of the cart down the ramp can be considered a component of the gravitational acceleration: a = g sinθ

Then we have an incline of angle  given by Equation 1:

. (1)
The acceleration of gravity, g, acts vertically downward, so the component of parallel to the incline – which is the acceleration of our cart – is given by Equation 2:
(2)

We see in Equation 2 that a graph of acceleration a as a function of sinθ should be linear with slope g. We will take data to plot such a graph and from its slope determine the value of g.

Setup
Gather the following materials:

· 2 m ramp

· Meter stick

· Lab Stand

· Ramp clamp

· Plastic Box with ULI, AC Adapter, and USB Cable

· Motion Sensor

· Magnetic Bumper

1. Connect the ULI to the computer via the USB cable and connect the AC adapter. Open Logger Pro 3.8.7.

2. Attach the ramp clamp to the lab stand and attach one end of the ramp.

3. Elevate one end of the track slightly using the vertical rod. Choose a value of h so that the angle of inclination stays less than about 8 degrees. (Use Equation 1 to verify).

· You can choose any two points along the track to serve as your L, but they must be the same two points for all your runs!

· Measure h by measuring the difference in the two heights of your two points.

4. Connect a motion sensor to the ULI and mount it on the elevated end of the track. The low end of the track should have a magnetic bumper installed on it (magnets face upward along the track).

Procedure
1. Choose at least five values of height h, to vary  over the range 1-8 degrees.

2. Record each value of h chosen, and then obtain a graph of velocity versus time for that value.

3. You have two options for collecting velocity data from the cart:

· Release from the elevated end of the track and let it accelerate to the lower end.

· Push the cart from the lower end of the track up the incline. Record data during its entire motion back to its starting point. This will take slightly more finesse, but the data will be better.

The motion sensor will not record accurate data for a cart closer than 40 cm (the limit of its near range). Do not let the cart collide with the end of the track!

4. Determine the acceleration for the cart by using the Linear Fit tool and highlighting the appropriate region of the velocity graph. Record the acceleration (the slope of the line of the velocity curve for the part of the curve where the cart is accelerating).

5. Repeat steps 3 and 4 two more times. Take the average of the three runs to get your acceleration.

6. Repeat steps 2 through 5 for the five values of your height.

Copy the image of ONE of your velocity graphs (with linear fit) for your lab write-up!

Data Analysis
1. Create a graph of acceleration versus the sinθ.

a. To plot a graph of acceleration versus height, we will need to enter our data into a new table. In order to do this, unplug the motion sensor from CH 1.

b. Create a new data table by selecting New Data Set from the Data menu. Enter the data for sinθ (X) and acceleration a (Y). The data points should automatically be entered into Logger Pro.

c. The column names in the data table can be renamed by double clicking on the title columns in the Data Table in the Data window. A box will pop up.

i. In the Column Definition tab, you can set the name, the short name, and the units for each column.

ii. In order to change the shape of the data points, double click on the second column and go to the Options tab. Under Point Symbols, select the shape you like.

d. Be sure to give an appropriate title to the graph (Right-click on the graph and select Graph Options) and the axes (see part b).

e. Create a linear fit of the graph by highlighting the data points on the graph and selecting the Linear Fit button.

f. Copy/Save an image of this graph (with the linear fit) for your write-up.

2.

Use the slope of the acceleration versus sinθ graph (either from the computer or from your lab notebook) to calculate your experimental value of g. Compare your experimentally measured value of g with the “accepted” value (the value many other experimenters have found for g at this location on Earth) by computing the percent difference between your experimental value and the accepted value of . A “good” value for the purposes of this lab is a percent difference of 5%.
Instructions for Lab Report

· Lab reports are due on the date specified on Canvas. Late lab reports will suffer a 10% per day penalty unless an extraordinary circumstance applies.

· Lab reports must be prepared on a word processor. They must be written in proper grammar. Graphs and tables should be included in-line or printed out, not pasted or taped into the report.

· Lab reports should be in past tense. You already did the experiment, not discussing an ongoing project. In addition, the report must follow the standard conventions of written English.

The lab report must include the following sections:

Headings: Title, your name, Lab partner name(s), date.

Purpose: Here you tell what you set out to investigate or measure in the lab. This section should ordinarily be brief and to the point (1-2 sentences).

Procedure: Here you specify what equipment you used and what you did in the experiment. You should use a figure to explain the setup. You may use the figures in the Lab Manual Word document, but use your own words, i.e. do not copy and paste text from the Lab Manual. You should not enumerate every single thing you did. Just explain clearly what you did to obtain your data, so that another person reading the report could understand what you did.

Data: Here you should present the data you obtained, usually in graphical or tabular form. You should have the following:

· A sample graph of one of the five runs you took, including the slope determined by Logger Pro.

· A table that shows data from the five good runs in which you graphed velocity versus time. For each run the table should show h in meters and the acceleration in m/s2 you obtained from the slope of the line for that run.

· A graph of acceleration in m/s2 versus h in meters for the five runs that includes a linear fit and shows the slope from which you obtain the value of g.

All graphs and tables must have titles and appropriate units for the quantities plotted or tabulated. All graphs and tables must have captions and references! (See Captions and References for further reference.)

Calculations: Here you show the calculations you made of the quantities you were trying to determine in the experiment. For this lab, this means:

· Calculation of the acceleration g due to gravity from your slope of the a vs h graph.

· Calculation of the percent difference between your measured value and the accepted value. Make sure all quantities are expressed in appropriate units.

All calculations must be explained; a few sentences will do here. (See the Captions and References section for further reference).

Discussion: Here you explain your results. Discuss quantitatively the possible sources of error in your experiment and, determine sources of uncertainty and error in your final experimental values. This should always be done, regardless if you achieved an “acceptable” value.

Conclusions: Here you tell how well you achieved your purpose. This section should be brief and to the point. For Lab 1, your conclusion should include the value of g that you found and its percent difference from the accepted value. If you like, you can combine this with the Discussion.

The lab will be graded using the following rubric:

Purpose: 2 pts

Procedure: 6 pts

Data/Calculations: 10 pts

Discussion/Conclusion: 7 pts

Captions and References

Figures, tables, diagrams, and equations are useful ways to display a lot of information. If used correctly, they are more effective at displaying information than a bunch of paragraphs. However, each of these must contain references and captions. Here is a brief description on the difference between the two.

References point the reader to a figure, table, or equation.

Captions are short descriptions of the figure or table. They often (but not always) repeat information in the lab report.

Here is an example of a figure being used with a proper reference and a caption:

“…Our basic setup of a mass/spring system is shown in Figure 1 below.

http://upload.wikimedia.org/wikipedia/ko/9/9f/Mass_spring.png

Figure 1 – A horizontal mass-spring system with mass m and spring constant k.

We take positive displacement x to the right…”

In this case, it is clear to the reader what figure is being referenced and where to find the figure. Usually, figures and tables are noted with a counter (i.e. Figure 2, Table 1). All figures, equations and tables must have captions and references, otherwise how would the reader know when to look for them?

Equations are referenced a little differently – instead of a caption, it is customary to indicate a number (or letter) to the side of the equation. Here is an example on how to correctly reference an equation:

“…We calculated the time t for the projectile to hit the ground using Equation 3:

(3)
where h is the initial height of the projectile and g is the magnitude of the acceleration due to gravity, 9.80 m/s2.”

Also note that the meaning of each of the variables is clearly explained. Do not assume your reader knows your variables or your notation – this is essential for clear communication!

A couple of notes before you get started.

· Equations can be hand-drawn. However, they must be neatly drawn/written and be on the same type of paper that the rest of the report is in (i.e. no lined paper).

· Figures, tables, and equations don’t need to be on the same page as the reference. They can be placed on another page or on the back of the report. In this case it is customary to indicate the page number where the figure is placed (e.g. “As seen in Table 1 (pg. 7), we measured the height of the launcher…”).

· If you are using Microsoft Word to type up your report, there is a handy equation editor included. Click on the “Insert” tab and then click the “Equation” button. The toolbar will change and you will be able to type out equations, insert special scripts and letters, etc. Let me know if you need help. (As always, you can hand write out equations. Just make them neat).

If you have any more questions, do not hesitate to ask!

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43. Continuous Adiabatic Heat Exchanger
Propane gas enters a continuous adiabatic heat exchanger’ at 40°C and 250 kPa and exits at 240°C superheated steam at 300°C and 5.0 bar enters the exchanger flowing counter currently to the propane and exits as a saturated liquid at the same pressure.
(a) Taking as a basis 100 mol of propane fed to the exchanger, draw and label a process flowchart. Include in your labeling the volume of propane fed (m3), the mass of steam fed (kg), and the volume of steam fed (m3).
(b) Calculate values of the labeled specific enthalpies in the following inlet—outlet enthalpy table for this process.
(c) Use an energy balance to calculate the required mass feed rate of the steam. Then calculate the volumetric feed ratio of the two streams (m3 steam fed/m3 propane fed). Assume ideal gas behavior for the propane but not the steam and recall that the exchanger is adiabatic.
(d) Calculate the heat transferred from the water to the propane (kJ/m3 propane fed). (Hint: Do an energy balance on either the water or the propane rather than on the entire heat exchanger.)
(e) Over a period of time, scale builds up on the heat transfer surface, resulting in a lower rate of heat transfer between the propane and the steam. What changes in the outlet streams would you expect to see as a result of the decreased heattransfer?