15-March-2017: Lab 5: Trajectories

Lab 5: Trajectories

Author: Tian Cih Jiao

Lab Partners: Weisheng Zhang, Kitarou

Date: March 15, 2017

Purpose of the lab: To use our understanding of projectile motion to predict the impact point of a ball on an inclined board.

Theory/Introduction: 
For the first part, we drop the ball from the high place then let the ball go following the channel then fall to the paper put on the floor, then the ball will create a point on the paper. We did this for 5 times for the part 1. Then we got the height and distance and also the angel of the ramp, then we calculate the result to predict the ideal result for the next part.
For the second part, the ball is dropped in the exact same pattern, but in this part, the ball is not falling to the ground but on a ramp wood next to the table with a new angel. There is a new paper put on the wood to record points created by ball. Then we got the result from this part which is the real result then we compare it with what we predicted in part 1.

Summary of apparatus/experimental procedure: Aluminum “v-channel”, steel ball, board, ring stand, clamp, paper, carbon paper

For Part 1 on the ground:

For  Part 2 on the wood:

Measured data:

For both parts:

The height of the table is measured to be 0.94 +/- 0.001m.

The angle of the channel is measured to be 12 +/- 2^o.

For the first part:

The distances from the edge the table are:

0.54, 0.543, 0.55, 0.555, 0.558 +/- 0.0001m

The average distance is (0.54 + 0.543 + 0.55 + 0.555 + 0.558) / 5 = 0.5492m.

For the second part:

The angle of the leaning wood is 48 +/- 2^o from the ground

The distances from the edge of the wood leaning towards the table are:

0.546, 0.564, 0.57, 0.576, 0.609 +/- 0.0001m

Calculated result(s)/Graph:
h is the height of the channel
V0x is the velocity where it launches off the table
d is the distance from the top of the wooden ramp to where the ball drops
Eventually we are calculating for the propagated uncertainty of d

Analysis:

In this lab, we measure the ball dropping location and the height, then we use equation to find the ideal result to compare with ours. There are still some uncertainty like friction of channel, air friction, and some measurement errors. 

Conclusion:

Our result compare to the calculated ideal result is very close. We did the dropping process for 5 times for each to make the uncertainty smaller.

13-March-2017: Lab 4: Modeling the fall of an object falling with air resistance

Lab 4: Modeling the fall of an object falling with air resistance

Author: Tian Cih Jiao

Lab Partners: Weisheng Zhang, Kitarou

Date: March 13, 2017

Purpose of the lab:

The purpose of the lab is to determine the relationship between air resistance force and speed. Based on the theoretical model we create, the falling of an object including air resistance is modeled by applying the mathematical model developed in the first part to predict the terminal velocity of various coffee filters.

Theory/Introduction:

Summary of apparatus/experimental procedure:

Measured data:

We capture video for coffee filter in build 13. Then we add points on the video then we get a position verse time graph.

The slope of this line is the velocity of the filter. Then we get five of the same process for all filters. Following picture is the third coffee filter's linear fit graph.

This lab is to find air resistance. So we should find a point of time, the gravitational force equals to air resistance. Then the velocity is the terminal velocity. By applying power fit (F = a * v^b), we get the following:

As a = 0.006143066 +/- 0.00009901, b = 1.987616 +/- 0.1687 (in this case, k = a, n = b), the mathematical model we developed was:

F = 0.006143066 * v^1.987616

Since air resistance force is related to the mass of the object, we measured the mass of 50 coffee filters in order to minimize the impact of propagated uncertainty.

let dt = 0.0001 so that v won't change too much. Then use the formula to find the terminal velocity for each case.

One of the case (6 filters):

The final predicted value is 2.9082054 which is very close to what we got in the LoggerPro which is 2.876. Then we do the same process for the other four times. Here is a summary of predicted and analyzed results:

Percent error = (Analyzed results – Predicted results) / Predicted results * 100%

Analysis: 

After comparing the result of real and predicted, we find that they are very close. There are many factors that would affect the errors. When we add points for the video, we might be not very accurate. And also the falling is not ideal free-fall.

Conclusion:

In this lab, we capture 5-6 times filter falling and add points for each video to get position verse time graphs. Then we get the lab terminal velocity. Then we use mathematical model to get ideal terminal velocity result which is very close to ours result.

8-March-2017 : Lab 6: Propagated uncertainty in measurements

Lab 6: Propagated uncertainty in measurements

Author: Tiancih Jiao

Lab Partners: Weisheng Zhang, Kitarou

Date: March 8, 2017

Purpose: Learn how to read vernier calipers and use it to measure the height and diameter, then find the propagated error.

Theory/Introduction:

In this lab, first we need to know how to read the vernier, then we use it to measure the height and diameter of each metal. Then we use the derivative to calculate the propagated error for my density measurements.

Summary of apparatus/experimental procedure: 

We used vernier to measure the height and diameter.

Professor showed how to read first:

Metal:

Measurement:

The calculation process:

Analysis: From the result of this lab, we can see the density of Al is about 2.713 to 2.771, and density of Zn is about 6.5973 to 6.8127, our result is close to acceptable value, but we also know that there is some human errors during measuring.

Conclusions: 

In this lab, we measured the height and diameter of two metals, then we use derivative to find out their densities. We can see the uncertain errors during this lab.

8-March-2017: Lab 3: Non-Constant acceleration problem/Activity

Lab 1: Finding a relationship between mass and period for an inertial balance

Author: Tian Cih Jiao

Lab Partners: Weisheng Zhang, Kitarou

Date: March 8, 2017

Purpose of the lab: The purpose of the lab is to find out how far an elephant goes before coming to rest with the given scenario. A 5000-kg elephant frictionless roller skates in going 25m/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 8000N 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 20kg/s) so that m(t) = 1500 kg – 20 kg/s * t (rocket).

Theory/Introduction:

We create a new Excel spreadsheet and input all the parameters and formulas into it. Then we can see the change of acceleration, velocity and distance.

Summary of apparatus/experimental procedure:

We use one groupmate's laptop doing the lab. He used excel to get the result of the lab.

Measured data: There is no measured data for this lab. Everything is carried out based on the known parameters in the scenario.

Calculated result(s)/Graph: 

Analysis:

After we put formula for every variables, we can just change the dt smaller to make the result more accurate. Then we just fill down the table so that we get the v=0 at rest.

Questions:

1. Compare the results you get from doing the problem analytically and doing it numerically.

By looking at the table whose dt is 1s, it makes sense since the results match almost perfectly.

2. How do you know when the time interval you chose for doing the integration is “small enough”? How would you tell if you didn’t have the analytical result to which you could compare your numerical result?

By solving the problem analytically, the integration is “small enough” when the time interval chosen is relatively small comparing to the theoretical “whole” time it takes. Without the analytical result, we can look at the velocity volume. If it quickly decreases to something below zero, then we can tell that the time interval chosen is too large.

3. Determine how far the elephant would go if its initial mass were 5500 kg, the rocket mass is still 1500 kg, but now the fuel burn rate is 40kg/s and the thrust force is 13000N.

By doing the integral from 0 to x, dv = 325ln(175-x)-325ln(175), v=v0+dv=25-325ln(175)+325ln(175-x)

By doing the integral for the distance.

We can use small number of time interval to fill down the table, then we get that the elephant will stop at 119 meters.

Conclusion:

In this lab, we get the result by setting equation of velocity and acceleration. Then we use excel to fill down the table to find the changing of variables.

1-March-2017 : Lab 2: Free Fall Lab – determination of g and some statistics for analyzing data

Lab 2: Free Fall Lab – determination of g and some statistics for analyzing data

Author: Tiancih Jiao

Lab Partners: Weisheng Zhang, Kitarou

Date: March 1, 2017

Purpose: 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/s^2

Theory/Introduction:

This lab is to demonstrate the motion of a freely falling body. When the free-fall body, held at the top by an electromagnet, is released, its fall is precisely recorded by a spark generator. We will pull a piece of paper tape to record the points. Then we use the data to make a distance-time graph and a velocity-time graph. We can use them to calculate acceleration.

The spark box is 60 Hz which is 60 points in one sec.

Summary of apparatus/experimental procedure:  

We use laptop's Excel to calculate ∆x, Mid-interval time and speed. Based on these data, graphs are plotted.

Deviations using mathematical equations:

∆x1 = x1 – x0 (for example, follow the pattern for the remaining ∆x)

Mid-interval time = corresponding time + 1/120 (s)

Mid-interval speed = ∆x / (1/60)

Measured data:

Calculated Data and graph:

Average deviation of the mean =

Standard deviation of the mean =

Analysis:

From the graph, we can see that it's not constantly increasing, which is caused by some random errors. It is because when we read the rule of these points, we might read some point bigger or smaller, so it is random error. Since there are more points of the slope, the derived acceleration of gravity may be larger than the exact value in that situation.

Questions:

Part 1: 

1. Show that, for constant acceleration, the velocity in the middle of a time interval is the same as the average velocity for that time interval.

The velocity in the middle of a time interval: V0 + a(T2 – T1)/2

The average velocity for that time interval: [V0 + (V0+ a(T2 – T1))] / 2

They are the same.

2. Describe how you can get the acceleration due to gravity from your velocity/time graph. Compare your result with the accepted value.

From the graph, we can get a from the equation: V= V0 + at.

Then we can calculate that a = 9.853 which is very close to g = 9.8.

3. Describe how you can get the acceleration due to gravity from your position/time graph. Compare your result with the accepted value.

From the position graph, we can get g from the equation: y = y0 + 1/2 * g * t^2,

Then we can get g = 9.327 which is less than 9.8.

Part 2:

1. What pattern (if any) is there the values of our values of g?

The first is closer than the second one because there is random error with second one. 

2. How does our average value compare with the accepted value of g?

We can get average g is 9.6256. We will probably do this experimental steps more times to get the result closer to 9.8.

3. What pattern (if any) is there in the class’ values of g?

It is close to 9.6, because air resistance and some other factors will decrease the accelerate.

4. What might account with any difference between the average value of your measurements and those of the class? Which of these are systematic errors? Which are random errors?

Reading on the rule will cause the random errors, air resistance can be the systematic errors.

5. Write a paragraph summarizing the point of this part of the lab. What were the key ideas? What were you supposed to get out of it?

In the lab, we will never get the most accurate data because there are too many invisible errors that we cannot avoid. What we should do is to use better method and repeat the experiment then get average result to find the final accurate data.

Conclusions:

For the first part, we use free-fall body, spark box and tape to record the points on the tape and make two graphs to find the g.

For the second part, we learn how to minimize the error and factors in experiment, and compare our result with errors to the accurate value of g.

27-Feb-2017: Finding a relationship between mass and period for an inertial balance

Lab 1: Finding a relationship between mass and period for an inertial balance

Author: Tiancih Jiao

Lab partners: Weisheng Zhang, Kitarou

Date: 2017/2/27

For this lab, we can measure the period of oscillation for a bunch of different masses, then we will use it to determine the unknown masses of many other objects.

Things to measure: The oscillation period of the inertial pendulum for a variety of known masses; The oscillation period of the inertial pendulum for a couple of unknown masses.

Theory/Introduction:

We are going to guess that the period is related to mass by the equation:

T = A(m + Mtray)^n

We have three unknowns--A, Mtray, and n. So we have to find them by the experiment.

Then we take the natural logarithm of both sides. Then we have:

lnT = n ln(m + Mtray) + lnA , which looks like y = mx + b

Y-intercept = lnA

X = ln(m + Mtray)

slope = n

We will try different values to get a straight line for Mtray.

Then we will come up with a range of A and n, and each value of Mtray will give us its own values for A and n.

Summary of our apparatus/experimental procedure:

We use a C-clamp to secure the inertial to the tabletop. Put a thin piece of masking tape on the end of the inertial balance just like the picture.

We use LabPro with a power adapter, USB cable, and plug adapter with laptop. Then we use them to measure and record the period with different mass on it.

Measured Data:

We enter these data into laptop, then we get a graph.

We guess 2 times for the Mtray, then we adjust the correlation to 0.9998, which is require to be from 0.9997-0.9999. The m = 0.8157, and the b = -6.175

Then we changed the value of Mtray to see the range of Slope and Y-intercept.

After we got the min and max of them, we can calculate the unknown items, so we use a pencil bag and an NDS as unknown items to test.

We use the n and lnA, and m is unknown, then we solve for m in the equation:

lnT = nln(m + Mtray) + lnA

Then we get the following table.

Analysis:
We measure and record data by changing the mass or the object, then we can get the range of the lnA and n. Then we can use the known lnA and n to calculate the unknown items mass. This is the lab purpose to find the relationship between the mass and the period.

Conclusion:
The purpose of the lab is to find the relationship between mass and period. So first we use laptop and C-clamp to record the period by increasing the mass. Then we find lnA and n by the given equation. After getting two unknown values of three, we can use these two values to find the rest one. So we can use the result to find out some other items' mass.