24 September 2014: Circular Motion and the Relationship Between Angular Speed and the Angle of a String

Purpose: To derive a relationship between angular speed (ω) and the angle of a string (θ) for a particular apparatus. 


The apparatus below rotates about a fixed point at some speed with the

22 September 2014: Centripetal Acceleration as a Function of Angular Speed

Purpose: To find centripetal acceleration as a function of the angular speed of the system.

15 September 2014: Coefficients of Friction

Purpose: To find the coefficients of kinetic and static friction on a horizontal and an angled surface.  

     To predict acceleration of a two mass system with one mass on a slope.

8 September 2014: Propagated Uncertainty

Purpose: Part 1: To calculate the propagated error in each of one's density measurements. Determine

     whether or not those measurements are within the experimental uncertainty of the accepted values.

     Part 2: Determination of a suspended unknown mass.


     The apparatuses below are a scale accurate to 0.1 grams and a caliper accurate to 0.01 cm.

     We used this equipment to measure the mass, the diameter, and the height of the three cylinders of 

brass, steel and copper. 

Below is partial calculations of density and its propagated error

The density of brass is 8.43g/cm^3 with p +/- dp 

Part II: Determination of an unknown suspended mass

     

      A labeled picture of the unknown suspended mss

Th

10 September 2014: Projectile Motion Lab

Purpose: To use one's understanding of projectile motion to predict the impact point of a ball on an

     inclined board.

Equipment:
     1. Aluminum "v-channels"
     2. Steel ball
     3. Board
     4. Ring stand
     5. Clamp
     6. Paper
     7. Carbon paper

     The apparatus was used to release the steel ball from a point on the sloped aluminum "v-channel"

so that it rolled down the first v-channel onto the second one, which ended at one of the edges of the

table, so that the ball would leave the table at some velocity and drop some distance away from the

edge of the table. The carbon paper along with the another piece of paper would be positioned so that

when the ball hit the floor it would hit the carbon paper, thus leaving a mark on the other piece of

paper, at the point of impact.

     We measured the height of the end of the horizontal track to be 88.5 cm. After dropping the ball 

the displacement from the edge of the table on the floor in the x direction found to be 59 cm after a 

series of five tests.

      From this we were able to calculate launch velocity by finding t in the equation h=-(1/2)*gt^2 as 

we know the height we found that t=0.425 seconds. With t and the known displacement we were able 

to find v in the equation ∆x=vt finding the velocity to be 1.388 m/s. 

     

     With this new information we were able to derive an equation that would allow us to determine 

how far the ball would strike a plank on an incline of some angle Lcosθ=vt. From this we find t to be 

t=Lcosθ/v. With the equation for height we find that h=1/2gt^2, which becomes Lsinθ=(1/2)gt^2. We 

then replace t with the equation for t that we found earlier Lsinθ=(1/2)g(Lcosθ/v)^2.     

     After placing the board down we found the angle between the board and the ground to be 51°. We 

then input this along with the other information that we know into the formula found above. 

Lsin(51)=(1/2)*9.8*(Lcos(51)/1.388)^2. Doing the math the prediction for L comes out to be 0.452m.  

Our setup with the plank of wood

     The experimental distance turned out to be d±σ = 0.460m this has a 1.8% error from the 

theoretical distance. Sources of error or uncertainty may come from an incorrect measurement of 

the plank or a different releasing height of the ball.

Summary

     We setup an apparatus that allowed us to test projectile motion. Comparing our theoretical value

based off the physics equations to the experimental value that we measured as a result of our

experiment. Finally, commenting on possible sources of error or uncertainty.

8 September 2014 Modeling Air Resistance

Purpose: Part 1: To determine the relationship between air resistance force and speed.
                Part 2: Model the fall of an object including air resistance.

Equipment used:
     1. 15 Coffee filters of the same size: to drop and measure air resistance
     2. A 2 meter stick
     3. A Macbook Pro with Logger Pro, Video Capture, and Microsoft Excel: To map and chart the
     data

     The two pictures below show two of five separate tests where we as a class, in our groups, in

dropped five sets of coffee filters from a balcony. The the sets consisted of 1, 2, 3, 4, and 5 coffee

filters respectively. Using video capture on the Macbook Pro we were able to record the fall of the

coffee filter(s). With Logger Pro we were able to map out the rate and acceleration at which the

coffee filter(s) fell. The dots represents the approximate location of the coffee filter(s) at some time.

The 2 meter stick was used to scale our distance during the fall of the coffee filter(s).

Pictures of Video Capture Analysis

     Below are the five position over time graphs that we found and their respective slopes.

     Above is our mg vs velocity graph as the set of 5 coffee filters worked best we shall use this below 

to compare our experimental data against our theoretical data. The slope becomes our k that is 

     While we were gathering our equipment for the lab we found that the coffee filters that we used 

were each roughly gram making mg=0.01 N. Using the slopes of ends of the position over time 

graphs we had the terminal velocity. 

   

  The sum of all forces for this lab was ∑F=ma=mg-R this give us an acceleration of a=(mg-R)/m. 

R is the force of air resistance. Using the model R=k*v^n where k is some constant and n is some 

power we were able to insert it into our formula for acceleration a=g-k(v^n)/m. When the

acceleration is 0 k becomes 0=9.8-k((2.771)/0.005) k comes

3 September 2014: Non-Constant Acceleration

Purpose: To solve a problem with a non constant acceleration

For this we used only Microsoft Excel 

      As a class, along with the Professor Wolf, we solved this problem. Following the steps on the

handout and those told to us by the professor.


      Based off the original instructions my group and I found the time, acceleration, the average

acceleration, the change in velocity, the velocity and the distance for the elephant. The first 35 points

are shown below when the interval of time is 0.1 seconds. Thanks to Excel the numbers and

calculations were easy to numerically integrate.

      Below is when the elephant's v=0 at some time in between 19.1 and 19.2 seconds in that time 

the elephant traveled about 243.8 meters. The time 19.1 and 19.2 seconds are highlighted.

    

 Here is how the problem is solved analytically. (From top to bottom). 

     

    

     When the time interval is 1s instead of 0.1s where v=0 at some time in between 19 and 20s 

these two times are highlighted creating a higher margin of uncertainty.

     When the time interval is 0.5 seconds instead of 0.1 seconds where v=0m/s at some time between 

19 and 19.5 seconds these two times are highlighted creating a higher margin of uncertainty than 

when the time interval is 0.1 seconds but less than when the time interval is 1 second.

Summary

     We solved a problem containing a non-constant acceleration as a class. Finding how long it would 

take to slow down an elephant on roller skates and how far that elephant went. The time when 

v=0m/s was 19.69 seconds and the distance was about 244 m numerically and I got 248.7 m 

analytically.

Conclusion

     Numerically I miss calculated at some point on the Excel sheet that I was uncertain how to fix as 

the analytically the math is correct. The time interval becomes small enough when there is a change 

of less than 1% as it fits into a three significant figures. 

27 August 2014 - Free Fall Lab and the Determination of g; Errors and Uncertainty

Free Fall Lab 

     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. 


The first step was to prepare the necessary equipment:
     1. A meterstick: to measure position
     2. A free-falling object
     3. Heavy tripod base with leveling screws    
     4. An electromagnet to release the object
     5. Tape and a spark generator: To mark the position as the object fell
     6. Computer: To calculate the data

    

     The picture above depicts a spark generator attached to a sturdy column with two wires, where a 

object is dropped from an electromagnet at the top. During the fall of the object the spark generator 

makes intervals on a spark-sensitive tape that records a permanent record of the fall. 

     

Above is a picture of multiple lengths of spark-sensitive tape.

     The picture above shows our length of spark-senstive tape. On this tape we marked 15 points out

and measured the total distance.


     We the proceeded to use Excel to create a chart where from the known time and distance came

up with five columns. These columns consisted of time, distance, the change of position in

between two points, the mid-interval time, and the mid-interval speed.

Time  (s)	Distance  (cm)	∆x (cm)	Mid-interval Time (s)	Mid-interval Speed (cm/s)
0	0	2.7	0.008333333	162
0.016666667	2.7	2.9	0.025	174
0.033333333	5.6	3.3	0.041666667	198
0.05	8.9	3.5	0.058333333	210
0.066666667	12.4	3.7	0.075	222
0.083333333	16.1	4	0.091666667	240
0.1	20.1	4.3	0.108333333	258
0.116666667	24.4	4.5	0.125	270
0.133333333	28.9	4.9	0.141666667	294
0.15	33.8	5	0.158333333	300
0.166666667	38.8	5.5	0.175	330
0.183333333	44.3	5.6	0.191666667	336
0.2	49.9	5.9	0.208333333	354
0.216666667	55.8	6.2	0.225	372
0.233333333	62	-	0.241666667	-

     In Excel we used formulas to come up with the change in distance, mid-interval velocity, and mid-

interval time.  From that information we graphed a chart of mid-interval velocity over mid-interval

time graph and as a result came up with the mid-interval acceleration as our slope. We also graphed a

position over time chart and came up with velocity as our slope. Both graphs are see below.

Graph of Mid-interval Velocity (cm/s) Over Mid-Interval Time (s)

Graph of Position (cm) Over Time (s)

      In class we combined the data that we got with the data of the other groups so that we would be 

able to find any deviation in our data results and find out any uncertainty that we had in the 

experiment. 

Group	g	Deviation	Abs Deviation	Deviation^2
1	961	5.711111111	5.711111111	32.61679012
2	936	-19.28888889	19.28888889	372.0612346
3	975	19.71111111	19.71111111	388.5279012
4	969.6	14.31111111	14.31111111	204.8079012
5	939	-16.28888889	16.28888889	265.3279012
6	975	19.71111111	19.71111111	388.5279012
7	949	-6.288888889	6.288888889	39.55012346
8	930	-25.28888889	25.28888889	639.5279012
9	963	7.711111111	7.711111111	59.46123457
Average:	955.2888889		14.92345679	16.29726933

     We found that as a class we had an absolute deviation of 0.149 m/s^2 of a fall due to gravity. As 

seen in the graph above.

Summary

     In class we attempted to validate the statement that with no other forces except gravity an object 

falls at a rate of 9.8 m/s^2. We dropped a object from an electromagnet that established a current 

between two wires on a column creating marks on a spark tape. From these marks we found the 

distance between 15 points and with the help of Excel the distance in between each of those points, 

the mid-interval time in between each point and the mid-interval speed. After this had been 

accomplished as a class we took down the all the groups information and averaged them out. We then 

found the average deviation of the classes' experiments from that average that we found for the 

groups cm/s^2 of gravity.

25 and 27 - August - 2014: Deriving a power law for an inertial pendulum

 Deriving a Power Law from an Inertial Pendulum

       In this lab we tried to find a relationship between mass and period for an inertial balance by 

comparing objects resistances to their changes in their motion.

Lab Equipment   

     The picture above contains the setup of an inertial balance along with weights, added later, and a

photo gate. These are used to find a relationship between the period of time that it takes an object of

some mass to pass through the photo gate twice.

From Data to Fulling Purpose

     After getting the data we plugged it into Logger Pro where we graphed the data. The objective was 

to find three unknowns in the equation T =A(m +Mtray)n so that we would be able to find the mass of 

two unknowns, given their period on the inertia pendulum. The three unknowns were A, Mtray, and 

n. We were able to do this by taking the natural log of each side lnT = n ln (m +Mtray) + ln A. We 

found, from plugging in the numbers, that the mass of the tray was in between 0.28kg and 0.33kg 

coming to an mean of 0.305kg. As this range for the mass of the tray created a beautiful straight line 

as seen by the very strong correlation between the plotted points. The slope of the line from the graph 

was n, the range was in between 0.6524 and 0.7133 having a mean of 0.68285. We found lnA to be 

-0.4168 and -0.4479 having the average of -.43235. Next from lnA we found the final unknown A to 

be in between 0.63897 and 0.65915 having the average of 0.649035  After we found that the three 

unknowns from the original formula we each separately plugged in several of the times we found 

from the known masses to confirmed that the formula and numbers closely matched up. As seen

below.

       In the lab we also found the periods of two unknown masses. Due to error on my part I do not 

have their times just an approximation based from the formulas. This is due to me being able to 

closely measure the masses of the two unknown masses, my calculator and a heavy duty carabiner 

having masses of approximately 0.292 kg and 0.105 kg from a scale in my home. The resulting 

periods were for my Unknown 1 with the mass of 0.292 kg being 0.4563 sec and for my Unknown 2 

with the mass of 0.105 kg being 0.353 sec. The math is seen below.

The Data Table is shown below

A screenshot from Logger Pro mass of the tray is 0.280 kg

     In this screenshot above the fit line is the lowest range at 0.280 kg of the mass of the tray creates 

a line with a very strong correlation. between the natural log of the mass of the tray and the mass 

of the object against the log of the period of motion.

Another screenshot from Logger Pro mass of the tray is 0.330 kg

   

  In this screenshot above the fit line is the lowest range at 0.330 kg of the mass of the tray creates a 

similar line with a very strong correlation. between the natural log of the mass of the tray and the 

mass of the object against the log of the period of motion.

Summary

     In this lab as a series of groups we attempted to find the relationship between mass and period by 

using an initial balance. As individual groups we took data using a piece of tape connected to an 

inertial pendulum that would pass through a photo gate and record data accurately. We then used

that data to confirm that the formula T =A(m +Mtray)n given to us was able to closely match the 

numbers we collected in our data. This allowed us to find two unknown masses from two objects 

of different masses given their periods of time passing through the photo gate. Though I had to 

find the time due to my misplacing of the information I took down.

Thank you for reading my lab blog.