Sep-12-2016: LAB 4 - Modeling the Fall of an Object with Air Resistance

LAB 4 - Modeling the Fall of an Object with Air Resistance

Tony Wu, Leslie Zhao, Isaiah Hernandez

September 12, 2016

Lab Goal: To determine the relationship between air resistance force and speed.

Theory/Introduction: In this lab we investigate the expectation that there is an air resistance force on a particular object that depends on the object's shape, speed, and material it is moving through.  We can model this expectation as a power law:

Fresistance=kvn

The k term takes into account the shape and area of the object. The variable n is a power constant.
Note that the force of resistance increases with time because the falling object speeds up over time.  We will find the terminal velocity of the falling objects.  The terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration.

Apparatus: In this lab, we used coffee filters as our falling object.  In order to avoid outdoor winds, we dropped 1 to 5 stacked coffee filters from a balcony.  We taped a meter stick to a black sheet as a background to track the coffee filters as they fell.  The meter stick is used as a reference to approximate the distance that the coffee filters fall per second.  

Experimental Procedure: We used Logger Pro to capture a video of each stack of coffee filters falling from the balcony.  Logger Pro allows us to track the speed of the coffee filters during its entire descent. Each coffee filter has a mass of 134.2 g ± 0.1 g.  

Data and Calculations: Below is the data for the free falls of 1, 2, 3, 4, and 5 coffee filters respectively
The graphs show Time vs. Position.  The slope of each graph is the approximate terminal velocity.

Free fall of 1 Coffee Filter


Free fall of 2 Coffee Filters


Free fall of 3 Coffee Filters


Free fall of 4 Coffee Filters


Free fall of 5 Coffee Filters


We can derive the values for k and n by taking the natural log of both sides of F = kvn equation, creating a y-intercept form line equation.  lnF=nlnv+lnkdfdfdfsfd
The slope of this equation will give us the value for n and the y intercept islnk.  

Below is the data shown from using Excel to estimate the terminal velocities.
Notice that the change in velocity rapidly approaches zero, signifying that the falling 
coffee filters hit terminal velocity!



Conclusions: In conclusion, this lab was a very good to free fall and terminal velocities.  Both Excel and Logger Pro are great tools to model and calculate the values for each free fall.  It is interesting to note that the coffee filters very rapidly approach terminal friction.

Sep-12-2016: LAB 3 - Non-Constant Acceleration

LAB 3 - Non-Constant Acceleration

Tony Wu, Leslie Zhao, Isaiah Hernandez

September 12, 2016

Lab Goal: To learn to use Excel to do integration for kinematics.

Theory/Introduction: Solving problems involving constant acceleration is a trivially easy task.  One simply needs to pick the right kinematics equations and then plug and chug.  On the other hand, problems that involve non-constant accelerations are more difficult to solve as they require calculus.  It is not difficult to use calculus to integrate the acceleration/velocity/position equations for simpler equations.  However, when the equations become significantly more complicated, it becomes very difficult to integrate multiple times.  In this lab, we learn to use Excel as a means to calculate the acceleration, velocity and positions at various times.  This allows us to simply plug in the data we recorded and from the Excel spreadsheet we can solve the problem numerically.

Apparatus: This lab did not use any apparatus.  We used Excel on our laptops.

Experimental Procedure: We are given the problem to solve analytically.

The Problem: A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on level ground.  At that point a 1500-kg rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion.  The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the m(t) = 1500 kg - 20 kg/s*t.  Find how far the elephant goes before coming to rest.

We first solve the problem analytically.  Newton's 2nd law gives us the acceleration of the elephant + rocket system as a function of time:

a(t)=Fnetm(t)=−8000N6500kg−20kgst=−400325−tms2
We can integrate the acceleration from 0 to t to find △v then derive an equation for v(t).In order to find out how far the elephant travels before coming to rest, we need to find the time at which the velocity of the elephant is zero. Now that we have the time, we can calculate the distance the elephant travels before stopping. We integrate the velocity from 0 to t to find △x and then derive an equation for x(t).  We now plug in the time t that the elephant stops into the distance equation and get the distance that the elephant stopped. We calculated the distance that the elephant would stop is 248.7 meters.

Now we are going to use Excel to solve the problem numerically. We set up the variables that we will plug into Excel: initial mass, initial velocity, burn rate, force in Newtons, and change in time.  We then have columns for time, a formula for acceleration, average acceleration, change in velocity, instantaneous velocity, average velocity, change in position and current position.


Data and Calculations: Below is the Excel spreadsheet.





Notice that in line 204, the velocity of the object reaches zero somewhere between 19.6 seconds and 19.7 seconds.  We able to estimate the distance at which the elephant stops, approximately 248.693 meters.

Conclusions: Analytically, we calculated that the distance at which the elephant stops moving was 248.7 meters.  Numerically, we reached a value of that is slightly greater 248.693 meters.  The numbers are practically equal.  In order to tell if the time interval we chose is small enough, we can pay close attention to how much each variable is changing by.  In conclusion, we can see the advantages of using Excel to solve these types of problems.  This method is especially useful for larger, more complicated equations that involve difficult integrations.

Given an initial mass of 5500 kg, a fuel burn rate of 40 kg/s and a thrust force of 13000 N, the elephant would travel a distance of 164 meters before coming to a stop.

Sep-7-2016: LAB 6 - Propagated Uncertainty in Measurements

LAB 6 - Propagated Uncertainty in Measurements

Tony Wu, Leslie Zhao, Isaiah Hernandez

September 7, 2016

Lab Goal: To calculate and understand the propagated error in each of our measurements. Determine whether or not our measurements are within the experimental uncertainty of the accepted values.

Theory/Introduction: Accuracy in measurements and calculations is an absolute must in the real world.  Structural engineers must work with a level of accuracy that is acceptable and safe.  With that said, 100% accuracy and precision is not possible.  Every measurement will have a certain degree of uncertainty.  These small degrees of uncertainty can propagate and drastically alter the final results.  In this lab we are learning to calculate the levels of uncertainty in order to explicitly state the range of possible values.  We learned how to use calipers to measure to an accuracy of 0.01 cm meters.

Apparatus: This lab only uses calipers for measuring length and a scale for measuring mass.

Experimental Procedure: We used calipers to measure the diameter and length of two metal cylinders. The diameter and length measurements have an uncertainty of ±0.01 cm.  We then measured the mass of each cylinder using a scale.  The mass measurements have an uncertainty of ±0.1 g.   Now that we have the mass, diameter, and length of each cylinder, we can derive the density of each. .

Data and Calculations:

 

Shown above are the calculations to derive the density of each cylinder. We use calculus in order to calculate the values for the propagated uncertainty. We take the formula for the density and take the partial derivatives of each variable. We then take the magnitude of the sum of all the variables.

Conclusions: Accuracy in measurements can never be 100% accurate. In order to take this into account, measurements must be given with an acceptable degree of uncertainty. Each little bit of uncertainty propagates to affect the final calculations. The important thing is to keep track of the limitations that every measurement tool has in order to calculate the propagated uncertainty.

Aug-31-2016: LAB 2 - Free Fall Determination of Acceleration due to Gravity

LAB 2 - Free Fall Determination of Acceleration due to Gravity

Tony Wu, Leslie Zhao, Isaiah Hernandez

August 31, 2016

Lab Goal: To examine the validity of the statement: In the absence of all other external forces except gravity, a falling body will accelerate at 9.8 m/s². The goal is to estimate the acceleration due to gravity and compare it to the accepted value.

Theory/Introduction: Using a well-known apparatus, we study the basic laws of motion in order to determine the acceleration due to gravity (g) of a freely falling body.  We predict that the acceleration due to gravity should be lower than 9.8 m/s² due to drag and/or wind resistance.  The apparatus we used releases an electromagnet.  The fall is precisely recorded by a spark generator.  The marks made at intervals on the spark-sensitive tape attached to the column give students a permanent record of the fall.  This data is then used to make a distance vs. time graph and a velocity vs. time graph.  Finally, the acceleration is calculated from velocity graph.  

Apparatus: The apparatus is 1.86 m tall.  It holds a 1.5 m length of spark-sensitive tape.  To set up the apparatus, pull a piece of paper tape tight between the vertical wire and the vertical post of the device. Clip it with a weight to keep the paper "tight".  The apparatus is now ready.

Experimental Procedure: We turned the dial hooked up to the electromagnet up a bit, then hung the wooden cylinder with the metal ring around it.  After turning on the power to the sparker, the spark leaves a dot on the paper.  We then turned off the electromagnet to initiate the fall of the object.  After the object has fallen, we turn off the power to the sparker thing and tear off the paper strip.  The marks on the paper strip correspond to an interval of 1/60th of a second. Now we take a two-meter stick and mark the position of each dot as measured from the first dot.

After we have all the dots marked and measured, we input the data into Excel.  In column A we have the time intervals and in column B we have the distance between each mark.  In column C we have the change in distance (delta x).  In column D we have the Mid-Interval Time, this gives the time for the middle of each 1/60ths interval. In Column E we have Mid-Interval Speed.

We then graphed all this data into a Marked Scatter chart.  We added a Linear Fit Trendline and also displayed an R-squared value on the chart.  The closer the value of R-squared to 1 is, the better the fit of the trendline. The Time vs. Distance graph is shown with a Polynomial Fit of order 2.

Data and Calculations: 

Graph 1: Mid-Interval Time vs. Mid-Interval Speed                           Graph 2: Time vs. Distance

The linear fit trendline equation, given slope-intercept form, best approximates the data recorded. The coefficient in front of the variable gives us the slope of the line which is the estimated value for the constant acceleration due to gravity.  The value we got was 9.63 m/s² which is lower than our expected value of 9.8 m/s².  We are also able to get the acceleration due to gravity from the Time vs. Distance graph.  We take the coefficient in front of the x² and multiply it by 2, 9.59 m/s².  This is also below our expected value of 9.8 m/s².  There are a number of reasons why the estimated values are different from the expected values, systematic errors such as air resistance and/or random errors such as user errors during the measuring of each spark mark on the tape.  Below are the values that each group got for the acceleration:

Group #1: 9.58 m/s²
Group #2: 9.63 m/s²
Group #3: 8.93 m/s²
Group #4: 9.57 m/s²
Group #5: 9.25 m/s²
Group #6: 9.55 m/s²
Group #7: 9.50 m/s²
Group #8: 9.85 m/s² 

Conclusions: 
In this lab, we assumed that air resistance is negligible.  Is it reasonable for us to make that assumption?   Every group but one had a lower estimated acceleration than the expected value.  This is likely due to the air resistance slowing down the freefall object.  As mentioned before, user error could also affect the data gathered.  We all measured the mark of each spark using a meter stick and eyeballing each measurement.  We can quantify how accurate our estimated value is by computing the absolute and relative difference to the expected value.

Absolute Difference - Experimental Value minus Accepted Value: 9.63 m/s² - 9.8 m/s² = -0.17 m/s²

Relative Difference - [(Experimental Value - Accepted Value)/Accepted Value] * 100%

[(9.63 m/s² - 9.8 m/s²) / 9.8m/s² ] * 100% = -1.73%

Our experimental value for the acceleration due to gravity is only 1.73% lower than the accepted value.