March 25, 2015 Centripetal Acceleration vs Angular Frequency

Objective: 
To determine the relationship between centripetal acceleration and angular speed.

The Set-Up:

Prepare an empty space, mount the accelerometer on the disk. The disk should be able to accelerate or decelerate smoothly while rotating. Set up the photogate on the side of the disk. Stick a piece of thin paper on the disk so that it can be detected when passing through the photogate.

Data Collection:
The equation for centripetal acceleration is a=rω^{2
Our goal is to calculate the radius (r) by doing an experiment to get the acceleration and angular frequency, and compare the value of r to the true value.
We measure the radius of the disk with a ruler, it is about 13.8-14 cm.
We begin the data collection by accelerating the disk using accelerometer and record the data through loggerpro.}
We have done five trials, each time with a different voltage from the accelerometer, which give us different acceleration.
We obtain the value of acceleration by looking at the acceleration vs time graph. We use the mean value as the acceleration.

Acceleration vs time graph

Next, in order to calculate omega, the angular frequency, we need the total time of rotation and the number of rotation. These information are found in another loggerpro data.

In this set of data, we are only interested in 1-16, the ones in red box because there we have the complete information. The total time is the difference between final time and starting time. In this particular data, we have 16.461-1.6716= 14.789 sec., and 8 rotations.

We input these data into excel to come up with the value of omega squared.

Acceleration vs omega squared graph

We obtain the value for omega squared. We plot the acceleration and omega squared, which gives us a graph above.The acceleration is the y axis and omega squared is x axis. The slope of this graph is the radius. We see that the slope is 0.1391 meter, which is also 13.91 centimeter. It is within our measured range of 13.8-14 cm.

Summary :
In this experiment, we seek to find the relationship between acceleration and angular frequency. We look at the equation a=rω², and set up an apparatus so that we can get both the acceleration and angular frequency. Then we calculate the value of radius and compare it to the true radius. The calculated radius is 13.91, and the true value is between 13.8-14 cm, we have achieve our goal.

Some error and uncertainty to this experiment would be we assume no friction between the accelerometer and the disk. In reality, we do not have a perfect equipment so there is friction. The friction could have slow down the the time to complete one revolution. In the same way, we assume no air resistance. Since the disk it rotating, there is air resistance involve, which slows the time to complete one revolution. There is also some human technique involve that could add more uncertainty and error to the experiment.

March 24, 2015 Trajectories Lab

Objective: Apply the knowledge of trajectory motion to predict the location as a ball falls from an inclined board.

The Set-Up:

We set up the above apparatus by putting together two metal v shaped bars. We take one bar and lift it up to a certain angle and stable the two metal bars with tapes. Then we test the apparatus with the ball to make sure the structure is stable.

Data Collection: 

we try five more times with a carbon paper placing at the spot where the ball falls, so that when the ball falls, the carbon paper immediately record down the exact location.

From the table to the dots is the horizontal distance. Since the table has rim, in order to accurately measure the distance, we hang a weight on a string when measuring. We measure the distance of each block dots from the hanging weight. Here is our data.

D1= 65 cm

D2= 64.5 cm

D3= 64.9 cm

D4= 64.8 cm

D5= 64.8 cm

We use 64.8 cm as our horizontal distance, with an uncertainty of +/- 0.2 cm. We also measure the vertical distance, the height of table, to be 94. 7 cm with an uncertainty of +/- 0.1 cm.

Putting the data together, we solve for the horizontal initial velocity. The horizontal velocity is 14.7 cm/s, which is 0.147 m/s.

Now we have the initial velocity of the ball, the big question that we still need to solve is if there is a board connecting to the exit of the structure, such that when the ball exit the structure, it falls on the board, what is the distance from the exit to the location where the ball falls?

Our job is to calculate the location, the impact point, then set up the board to confirm our result.

We measure the angle of the board to be 48.7 degree.

Here is our calculation

The distance we find is 76.1 cm or 0.761 m.

Frontal view of the apparatus with the board

Here is the experimental result.

The scale is in metric system. The experimental result is about 77.5 cm or 0.775 m.

Now we use the data to calculate the uncertainty in this result. To find the uncertainty of the distance, there are a few components to consider. These components are the angle of the board, the gravity, and the initial velocity, which includes the heights of the table and the horizontal distance from the rim of the table to the dots. These components are obtained through measuring, thus, there are some uncertainty involve. We take the partial derivative of each component. The first picture below shows the process of getting the partial derivative of initial velocity. The second picture shows the process of getting the partial derivative of the angle and the initial velocity. Gravity is not taken into account because the value of gravity, 9.81, is accurate. After much derivation, we find the uncertainty value for distance to be 0.775m+/- 0.004m

Finally, we also calculate the percent error. The percent error of this distance is about 1.84%.

Summary:

In this trajectory lab, our goal is to predict the exact location of the ball as it exits the track and falls onto the board. We begin by building the track. We did a few trial runs to see where the ball would land without the board. We mark our result with carbon paper then use it to measure the horizontal distance from the rim of the table. Then, we measure the height of the table to be our vertical distance. By having these few pieces of information, we calculate the initial velocity of the ball.

We set up the board and measure the angle. We use the initial velocity from previous calculation, the angle between the board and the ground, and some knowledge of trigonometry and trajectory, to find the distance from the rim of the table to the impact location. We run the experiment; the experimental result is not far from our calculation!

Finally, we calculate the uncertainty of this distance because some components are obtained through measuring. We found the distance with uncertainty to be 0.775m+/- 0.004m. And the percent error is 1.84%.

March 16/ March 18 Modeling Friction Forces

Objective: Five mini experiments are included in this modeling friction force lab. The five experiments examine different aspects of friction using different apparatuses.

Part 1: Static Friction
The Set-Up:

A wooden block sits on the surface of the metal track. The block is connected by a string to an empty Styrofoam cup. Both the cup and the block are not moving because the frictional force is greater than the tension force from the cup. The metal track is placed so that the cup is hanging in the air. In this way, the notable forces that acting on the cup would be tension from the string and the weight of the cup and water. We fill the cup with a little water at a time until the block begins to slide. Once the block slides, we weight the mass of the water. We also record the mass of the block.

Next, repeat this process with more blocks. We use four blocks in total, and we record the mass of the water and the mass of the blocks.

Data Collection:

The following is a table showing the mass of the block(s) and the mass of cup and water

Number of blocks	Mass of the block(s) in kg	Mass of cup and water in kg
1 block	0.1307	0.0321
2 blocks	0.2557	0.0429
3 blocks	0.3800	0.0852
4 blocks	0.6179	0.1037

Here is another table showing the normal force and frictional force of this set up. The normal force is the weight of the block(s). The maximum static friction force is the weight of the cup and water.

Number of blocks	Normal Force (N)	Friction Force (N)
1	1.28	0.31
2	2.51	0.42
3	3.72	0.84
4	6.06	1.02

Normal Force vs Friction Force Graph

Here is a graph taking normal force as x-axis and maximum static friction force as y-axis. The slope of the graph is the static friction between the block and the surface of the metal track. The value we obtain is µs= 0.23.

This is a calculation assuming the ideal situation. It also shows the calculation to get the coefficient of static friction, µs, between the block and the table. Here, the µs =0.24, compares to µs=0.23 using loggerpro.

Part 2: Kinetic Friction

The Set-Up:

We examine the kinetic friction by pulling a wooden block using a force senor. The wooden block has a red felt on its bottom and it is connected to a string. The other side of the string is connected to a force sensor. The data are recorded using loggerpro. A person pulls the block through the force sensor horizontally; the block is moves at a constant speed.  Then we can find the average force on loggerpro. In the same way, we repeat the process using two blocks, then three, until we get to four.

Data Collection:

Forces and time graph. These are the four trails records by loggerpro

The force values we get are:

Number of block(s)	The Pulling Force (N)
1	0.3415
2	0.6301
3	0.8197
4	1.196

The force and time graph for one block (first trial)

According to the graph, the average force for one block is 0.3415 N. The weight of the block(s) is the same as in part 1. We use the formula

F _pull = µ_k N

This is the table of normal force from part 1, we use the same blocks and the same order in this second experiment.

Number of blocks	Normal Force (N)
1	1.28
2	2.51
3	3.72
4	6.06

The force 0.3415 newton is the force for 1 block.

F pull = µk N

0.3415= µk (1.28)

µk=0.27

Please note that normally, the coefficient of static friction is greater than the coefficient of kinetic friction. Here we have µk= 0.27 and µs is 0.24 from the last experiment.

Part 3: Static Friction from a Sloped Surface

The Set-Up:

In this experiment, one block and an angle measurement device is place on the metal track. We tilt one end of the track until the block begins to move. Our goal here is to get the angle so that we can calculate the coefficient of static friction between the block and the metal track.

Data Collection:

The angle we get is 14 degree. 

We use Newton’s Second Law, F= ma, to calculate the coefficient of static friction. The coefficient of static friction is 0.24, just like part 1!

Part 4: Kinetic Friction from Sliding a Block Down and Incline

The Set-Up:

Here the metal track is tilted to a certain angle. The motion sensor is mounted on the higher end of the metal track. A block is to be let go a short distance away from the motion sensor. The block would accelerate downward. We use loggerpro to record the acceleration data. We want to find the coefficient of kinetic friction between the block and the surface of the metal track.

Data Collection:

The mass of the block is 0.1341 kg.

The incline is 25.4 degree.

Time vs Velocity graph, the slope is the acceleration

According to loggerpro, the acceleration of the block is 1.854 m/s^2.

Here is the calculation using F= ma.

The coefficient of kinetic friction of the metal track and the block is 0.266.

Part 5: Predicting the Acceleration of A Two Mass System

The Set-Up: 

The block is on the metal track, a string attachment the block to a weight. The weight is hanging freely in the air. The motion sensor is on the other side of the block to record the acceleration of the block as the weight falls.

Data Collection:

The goal of this experiment is to derive the acceleration of the block using the coefficient of kinetic friction we get from last experiment, µk=0.266. Then we check with loggerpro to see how accurate we are. 

From the calculation, we get an acceleration equals to 0.836 m/s².

Here is what the loggerpro records.

Velocity vs Time graph. The slope is the acceleration.

The loggerpro estimates that the acceleration is 0.5571 m/s².

It is different from our calculation result because loggerpro records the actual experiment instead of the ideal situation we assume in our calculation. During the experiment, some things that could affect the acceleration of the block may be the uneven surface of the metal track, human technique and air resistance

March 2, 2015 Free Fall Lab

Objective: Determine the acceleration of a free fall body is 9.8m/s^2 under no external force except gravity.

The Set-Up:

The Spark Generator

The apparatus for this experiment is the spark generator. The spark generator has a cylinder on the top, when the spark generator is turned on, the cylinder falls, as it falls, it leaves mark every 1/60th second. A long strip of paper is attach to the spark generator to record the spark that the spark generator generates.

Data Collection: 

The strip of paper with spark marks

For convenient reason, we are given the strip that already had spark marks on it. We measure the marks on the strip using meter stick, we measure with the unit of centimeter. This is our data.

Mark No.	Distance (cm)
1	1.4
2	3.1
3	4.9
4	7
5	9.3
6	11.9
7	14.7
8	17.8
9	21.3
10	24.9
11	28.8
12	33
13	37.5
14	42.2
15	27.2
16	52.4
17	58
18	63.8
19	68.7
20	76.1
21	82.6

Each mark represent 1/60th second. We input the data into the Microsoft Excel spread sheet. 

After we find the distance between each spark on the paper, 1/60th second apart, we input the formula into the spread sheet. Then we get a third column that has the distance per 1/60th second. This third column is the difference of distance.

Next, we make the fourth column titled "mid-interval time" by taking the original time plus 1/120. This gives the time for the middle of each 1/60 interval. Lastly, we produce the fifth column called "mid-interval speed" by diving the change in distance (∆x) by (∆t), 1/60 s.

We use the data to make two graphs. The first one is Mid-Interval Time vs Mid-Interval Speed and the other is Time vs Position graph.

Velocity/Time graph

position/ time graph

Questions/ Analysis:
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.

2. Describe how you can get the acceleration due to gravity from your velocity/ time graph. Compare your result with the accepted value.
The equation for the velocity/ time graph is y=939.74x+71.545. The figure 939.74x cm/sec^2 is the slope of this graph, which the acceleration, in this case, it is the gravity. 939.74 is measure in centimeter per second squared, when it is convert to meter, we get 9.4 m/sec^2. This is 5% less from the true value of gravity, 9.8 m/sec^2.
3. Describe how you can get the acceleration due to gravity from your position/ time graph. Compare your result with the accepted value.
The equation for the position/ time graph is y= 474.14x^2 + 69.661x +0.1514. When we take the derivative, we get y= 948.28x + 69.661. The gravity value we get is 948.28 cm/sec^2, which equals to 9.5 m/sec^2. This value is about 3 % less from the true value of gravity, 9.8 m/sec^2

Conclusions:
Assumptions of this lab
1. No friction (If there was friction, it would make the values smaller)
2. The sparks are exactly 1/60th second apart

Errors and Uncertainty



Toward the end of the lab, we take the gravity value from each group and analyze them. We make a excel spreadsheet to show the deviation from the mean. The way to get the deviation from the mean is by find the difference between the experimental values and the mean value. Squared the value and add every values together. Divide the sum of squared deviations by the number of data you have, in our case is 9. Then you take the square root of this number. Our standard deviation of the mean is 20.1.

According to the experimental data, the gravity is between 916cm/s^2-996cm/s^2, this is not a good experiment to find out the true value of gravity.

1.What pattern is there in the values of our values of g?
Our values of gravity are 9.4 m/s^2 and 9.5 m/s^2. The pattern is that they are close to each other.
2. How does our average value compare with the accepted values of g?
Our average value is 9.56 m/s^2, it is lower than the accepted value of gravity, 9.8 m/s^2.
3. What pattern is there in the class' values of g?
Class' values of gravity are smaller than the actual value of gravity.
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?
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?
This experiment has a system error which reflects in the class's average value of gravity to be lower than 9.81. This system error is the assumptions that we make for the spark generator. We assume that there is no friction nor air resistance. They are system error because no matter how careful our measurement or calculation is, the values are always going to be inaccurate, the problem is the whole system, not individual technique.
The random error is reflect on individual technique. The gravity value of my group is 9.4 m/s^2. This value is toward the lower end of the class data. It might due to the technique of the operator who made the strip, it might be because of the condition of the spark generator at that moment, it might be we made some small mistake when measuring the marks. These factors are random, thus they are called random error. 





March 11, 2015 Fall of an Object and Air Resistance Lab

Objective:
To find the relationship between the air resistance force and the speed of a falling object.

The Set-Up:

The photo of the balcony taken on the ground level

In Mt.SAC, building 13, the technology building, there is an indoor balcony. The advantage of indoor is that there is less unexpected force, such as wind, that could greatly affect our experiment. According to the instruction, some people stand on the balcony and drop the coffee filter. The way to drop the coffee filter is to hold it with two palms, and drop it gently, we want to make sure the coffee filter is stable while lowering.

Brown and white coffee filters

Our group choose the brown coffee filter to be the object because it is more visible than the white. We begin by lowering one coffee filter, we record by using video from the laptop, then we use loggerpro to help us analyze different points when the coffee filter is falling. We use a meter stick as a reference point for loggerpro. After we record the falling of one coffee filter, we add one more and did a recording with two coffee filters. We repeat the process until we have the video of five coffee filters falling.

Data Collection:

Velocity vs Time graph for 5 coffee filters

After recording five videos, we use loggerpro to analyze the videos. The results we get are five velocity vs time graph. The many dots on the graph represent different position of the coffee filter at different time. We choose the dots that are toward the end to analyze because they we are trying to find the terminal velocity and they are the ones that seems more consistent.  We use the linear fit option and get the slope of the graph; the slope is the terminal velocity.

Number of coffee filter (s)	Terminal Velocity (m/s)
1	0.9762
2	1.314
3	1.421
4	1.713
5	1.828

Next, we begin our calculation with an equation representing the air resistant force.

F= K * V^n

A few things to highlight of this equation:

·        ** The air resistant force (F) and v (terminal velocity) are proportional just like in the data table, the number of coffee filter is proportional to the terminal velocity

·         **The data from the table is a curved graph, it must involve some kind of exponent, therefore, in the equation v is to a power of n

·         **K represents a certain shape function

We can determine the values for k and n by plotting the data.

Distance and velocity graph

K=0.008846

n=2.678

Finally, we gather all the pieces of information and make a model according to Newton’s Second Law, F=ma.

Since this differential equation would be difficult to solve, we are going to use excel spreadsheet.

First we make eight columns and named them as t, delta v, v, a, delta x, and x. We set the time very small, such as 0.01 second so we can get a more accurate result.Under delta v, we input the formula, a1* delta t, because the acceleration times the change in time gives you the change in velocity. This change in velocity plus the initial velocity give you the final velocity, which is v. Next, gravity (9.8)- k/m* v^n gives us acceleration. The change in position is the average velocity times time. Lastly, final position is the sum of the initial positon and change in position.

Our goal is to look for a velocity when it becomes constant. This constant velocity is the terminal velocity of our experiment. 

Conclusion:

The excel that we use to derive the constant velocity

The terminal velocity is 1.842 m/s. This spreadsheet is calculated based on the mass of five coffee filters

Our experimental value of five coffee filters is 1.828 m/s, which is not far from our data!

The reason for this discrepancy might due to that fact that there is some other forces beside air resistance and gravity, such as when someone walks by or when someone opens the door, there could some random wind flowing around, which could affect the velocity. Other human caused errors could be the way we drop the coffee filter, though we try our best to keep it stable, the coffee filter could still fall on a random shape, which could influence the shape constant (K). Lastly, when we plot the point, we takes out a point, the velocity of three coffee filters, 1.421 m/s, because it is greatly out of the way. If we keep the third point, it could drastically change our result.

The graph with the third point that we taken out