Phys4AS17cceron

Sunday, April 23, 2017

April 03, 2017 Lab 9: Centripetal Force with a Motor

Lab 9: Centripetal Force with a Motor

Chris Ceron, Amy Chung, John Choi

Goal of Lab

Use apparatus to establish a relationship between $\theta$ and $\omega$

Apparatus and measurements used to establish the relationship between $\theta$ and $\omega$

Theory/Introduction

We know there exists a relationship between centripetal force and angular speed ( $\omega$ ):

$F_{c} = ma_{c} = mr\omega^2$

$r$ = radius

$\omega$ = angular speed

Through a free body diagram, we can establish a relationship of the different forces acting on our apparatus.

$\sum F_{x} = Tsin\theta = ma_{c}$

$\sum F_{y} = Tcos\theta - mg = 0 \Rightarrow Tcos\theta = mg$

We called the height from the ground to the top of the string H, and the height from the ground to the mass h. Using this, we can find the vertical length of the string as it is rotating, and allows us to solve for $\theta$ .

$cos\theta = \frac{H-h}{L}$ $\theta = cos^{-1}(\frac{H-h}{L})$

We can also derive for the angular speed using our sum of forces. We solved for T through the y-component of the forces and plugged it into T for the x-component:

$\sum F_{y} = Tcos\theta = mg \Rightarrow T = \frac{mg}{cos\theta}$

$\sum F_{x} = Tsin\theta = mr\omega^2 \Rightarrow \omega = \sqrt{\frac{Tsin\theta}{mr}}$

$\omega = \sqrt{\frac{mgsin\theta}{mrcos\theta}} = \sqrt{\frac{gtan\theta}{r}}$

r is the sum of the length of the top rod (R), and the x component of the string as it is rotating (Lsin $\theta$ ).

$\omega = \sqrt{\frac{gtan\theta}{R+Lsin\theta}}$

Procedure

Our apparatus is composed of a tripod with a vertical rod on top. Attached to the top of the rod is another rod attached horizontally. We attached a hanging mass to the end of the horizontal rod using a lightweight string. A motor attached to the tripod and rotates the rods and various speeds.

We used our apparatus for five different cases, each case with a different speed. For each case we recorded the amount of time it takes for the hanging mass to rotate ten times. For each case we also record the height (h) between the ground and the hanging mass by using a stand with a piece of paper attached to it. We can adjust the height of the paper until the hanging mass makes contact, then record the height between the ground and the paper.

Data:

The height from the ground to the top rod (H), the length of the rod (R), and the length of the string (L) are all constant in each case.

Height	Length	Radius
1.82 ± 0.001 m	1.59 ± 0.001m	0.75 ± 0.001m

Data for our five trial runs:

	height (m)	time for 10 rev (s)
1	0.36± 0.001 m	33.41
2	0.655± 0.001 m	27.37
3	0.87± 0.001 m	24.22
4	1.132± 0.001 m	19.92
5	1.36± 0.001 m	15.65

Calculated Data

In this lab, we found $\omega$ two ways. The first was mentioned in the theory/introduction section:

$\omega = \sqrt{\frac{gtan\theta}{R+Lsin\theta}}$

The second method used was to compare the time it took for one revolution:

$\omega = \frac{2\pi}{T}$

2 $\pi$ = one revolution
T = time period to complete one revolution

	θ (degrees)	ω (using angle) (rad/s)	ω (using period) (rad/s)
1	23.3	1.749505216	1.880630143
2	42.9	2.229344963	2.295646806
3	53.3	2.548188044	2.594213587
4	64.4	3.060365349	3.154209492
5	73.1	3.768444183	4.014814893

We used both calculations of ω, plotted them on a graph, and applied a linear fit.

Conclusion:

The two ω calculated should be identical. As shown in the graph above, the slope of the graph is 1.045, which is reasonably close to our expected value of 1. The uncertainty value of our slope is +/- .01039 or 0.994%. Our biggest source of error was recording the amount of time it took the hanging mass to revolve ten times. We recorded this measurement by using an application on our phone and timing when the mass approximately revolves ten times. Overall, we received a high correlation between both ω calculated. We can conclude that there is a relationship between $\theta$ and ω.

Tuesday, April 11, 2017

March 29, 2017 Lab 8: Centripetal Acceleration vs. Angular Frequency

Lab 8: Centripetal Acceleration vs. Angular Frequency

Chris Ceron, Amy Chung, John Choi

Purpose

To determine the relationship between centripetal force and angular speed by using an apparatus that allows for multiple fixed variable scenarios.

Apparatus used to find the relationship

Theory/Introduction

For an object moving in uniform circular motion, the direction of acceleration is towards the center. The force that corresponds to this acceleration is called the centripetal force:

$\sum F = ma_{c}$

The sum of the forces in our lab is shown through the following free body diagram of our apparatus

Through this free body diagram, we determined the sum of the forces in both the x and y direction

$\sum F_{y} = N - mg = 0 \Rightarrow N = mg$

$\sum F_{x} = T = ma_{c} \Rightarrow T = mr\omega^2$

omega can be represented in terms of period per each rotation:

$\omega = \frac{2\pi}{T}$

The sum of the forces in the x direction is a linear equation from which isolate a certain variable and keep the other two constant. Gathering multiple test cases allows for us to graph the data and apply a linear fit to find the isolated variable.

Experimental Procedure

Our apparatus is composed of a wooden circular disc with a pole protruding from the center. The disc sits on top of a motor that allows for the disc to rotate in a uniform circular motion. The motor was connected to a power supply that would allow for us to change the speed of rotation. We attached a force sensor to the pole, and attached a mass to the sensor using string. At the end of the disc was a piece of tape that would pass through a photogate after each rotation. This allowed for us to find the period between each rotation.

We allowed the disc to reach a constant speed before and would use LoggerPro to record the force. The force would fluctuate as the disc rotated, so we used the mean value.

The graph of force vs. time

The graph recorded for period of each rotation

The experiment was conducted several times, each time keeping a variable constant for a number of tries. The three variables we manipulating were mass, the radius, or omega. We would manipulate mass by changing the mass tied to the force sensor, the radius by either increasing or decreasing the length of the string that attached the mass to the force sensor, and omega by adjusting (increasing or decreasing) the power of the motor. Using all the data gathered, we could plot graphs for each variable we kept constant and apply a linear fit and find the correlation between the points plotted.

Data recorded through apparatus:

trial #	rotations	time initial (s)	time final (s)	Δtime (s)	radius (cm)	mass (kg)	Average Force (N)
1	10	1.428	14.769	13.341	22.86	0.2	1.2
2	10	1.728	14.879	13.151	29.21	0.2	1.47
3	5	13.36	20.11	6.75	34.29	0.2	1.691
4	10	1.57	15.6	14.03	46.99	0.2	1.981
5	10	2.42	17.44	15.02	59.44	0.2	2.198
6	10	2.68	17.02	14.34	46.99	0.1	1.052
7	10	1.451	14.97	13.519	46.99	0.05	0.57
8	10	2.35	16.46	14.11	46.99	0.3	2.801
9	10	1.24	11.48	10.24	46.99	0.3	5.393
10	10	1.12	10.75	9.63	46.99	0.3	5.94
11	10	1.46	10.6	9.14	46.99	0.3	6.434

Tuesday, April 4, 2017

March 22, 2017: Modeling Frictional Forces

Lab 7: Modeling Frictional Forces

Chris Ceron, Amy Chung, John Choi

Purpose

To model static and kinetic frictional forces through five different experiments by using different angled surfaces.

Theory/Introduction

Static and Kinetic Forces
source: http://physics-ref.blogspot.com/2014/11/physics-9702-doubts-help-page-6.html

1 - Static Friction (flat surface)
Static friction is the the frictional force acting between two objects as they are not moving. As shown in the figure above, the magnitude of the static frictional force equals the magnitude of the opposing force until the object begins to move. Once an object moves, a constant kinetic friction between both objects applies.

The maximum point of static friction can be modeled mathematically as:
$\mu_{s}N = f_{static \:\: max} \Rightarrow \mu_{s} = \frac{f_{static \:\: max}}{N}$
N - the normal force the surface exerts on the object

The coefficient of static friction, $\mu _{s}$ , equals the maximum value of static friction between the two surfaces divided by the normal force.

The force of static friction can be modeled as:

$f_{static} \leq \mu _{s}N$

where the static friction force can range from 0 to the maximum value of $\mu _{s}N$ .

Free body diagram for the set up of experiment one

M (wooden block on flat surface):

$\sum F_{x} = T_{1} -f_{s} = 0 \Rightarrow f_{s} = T_{1}$

$\sum F_{y} = N-Mg = 0 \Rightarrow N = Mg$

m (hanging mass):

$\sum F_{y} = T_{2} - mg = 0 \Rightarrow T_{2} = Mg$

$T_{1} = T_{2} \Rightarrow f_{s} = mg$

We modeled this free body diagram by attaching a wooden block to hanging mass using string and a friction-less pulley system. We placed the linoleum surface of the wooden block on a wooden board, and increases the hanging mass in small increments (increments of 5 grams) until the wooden block began to move. We used the total of the hanging mass to calculate the maximum static friction between the block and the board.

We repeated the process four times, each time adding an increment of 200 grams to the wooden block. Using the data recorded from the four cases, we graphed the normal force of the block as y and the force of static friction as x to model the equation of a line. Then we used a linear fit to find the slope, which is now the coefficient of static friction:

$y = mx + b \leftrightarrow f_{s} = \mu_{s}N$

The slope given by LoggerPro shows a value of 0.400.

2 - Kinetic Friction (on a flat surface)

Once the object begins to move, we see that kinetic friction remains constant and does not depend on the speed of the force. We model kinetic friction as:

$\mu_{k}N = f_{k} \Rightarrow \mu_{k} = \frac{f_{k}}{N}$

To calculate kinetic friction, we tied a force sensor to our wooden block using a string. We first calibrated the sensor by a 1-kg hanging mass, which is 9.8N due to gravity. We used LoggerPro to collect the data needed as we pulled the force sensor and block at a constant speed. Since our acceleration is zero, the force applied would be equal to the force of kinetic friction. Similar to the first experiment, we recorded four different cases, each with an increment of 200 grams to the wooden block.

We graphed the normal force vs. the mean of the force obtained by the force sensor found the coefficient of kinetic friction by finding the slope of the line.

Our slope value obtained, the coefficient of kinetic friction, was 0.2769 +/- 0.0048.

3 - Static Friction from a Sloped Surface

Static friction on a sloped surface reaches is maximum when the x component of the forces are big enough to move the object. To find the maximum static friction of the block, we slowly increases the slope of the wooden board until the block began to move. Through a free body diagram, we found our x and y components of the forces acting on the block:

$\sum F_{x} = mgcos\theta - f_{s} = 0 \Rightarrow f_{s} = mgcos\theta$

$\sum F_{y} = mgsin\theta - N = 0 \Rightarrow N = mgsin\theta$

We used a phone application to find the slope of the wooden panel and recorded the slope when the wooden block began to move. Using the angle, we were able to calculate the static friction. The angle recorded was 33 degrees.

$\mu_{s} = \frac{mgcos\theta}{mgsin\theta} = \frac{sin(33)}{cos(33)} = tan(33) = 0.65$

4 - Kinetic Friction from Sliding a Block Down an Incline

To measure the kinetic friction of the block, we held the wooden board at a slope that would cause the block to accelerate down the board. The acceleration of the block would allow for us to calculate kinetic friction. We placed a motion sensor on the top of the board and recorded the block's position vs. time, which would allow us to then calculate the acceleration of the block. We also used our phone application to record the slope of the wooden board.

The acceleration we obtained through LoggerPro is 4.071 m/s^2. The angle we used was 33 degrees. Our calculation for the coefficient of kinetic friction was:

$\sum F_{x} = mgsin\theta - f_{k} = ma$
$\sum F_{y} = N - mgcos\theta = 0 \Rightarrow N = mgcos\theta$
$\mu_{k} = \frac{mgsin\theta - ma}{mgcos\theta} = \frac{gsin\theta-ma}{gcos\theta} = 0.154$

5 - Predicting the Acceleration of a Two-Mass System

In a two-mass system (similar system to the one used in experiment one), we can add a hanging mass heavy enough to accelerate the wooden block. We attached a motion sensor to the opposite end of the table and measured the position of the block vs. time as it was being accelerated by the hanging mass.

We acceleration value obtained was 1.5 m/s^2. The hanging mass we used was 95 grams. Through our free body diagram, we obtained the following x and y component of the forces:

M:

$\sum F_{x} = T_{1} - f_{k} = Ma$
$\sum F_{y} = N - Mg = 0 \Rightarrow N = Mg$

m:

$\sum F_{y} = mg - T_{2} = ma$

Assuming T1 is equal to T2, and using the .183kg as mass of the block, and .095 kg for the hanging mass, we can calculate the coefficient of kinetic friction.
$\mu_{k} = \frac{mg - ma - Ma}{Mg} = \frac{0.095(9.8-1.5) - 0.183(1.5)}{0.183*9.8} = 0.287$

Conclusion

There were several discrepancies in the data that caused our calculated kinetic and static friction to widely vary in between experiments. These discrepancies were given before the lab, as we knew that each repetition of the procedure would come out with different data and calculations.

Gathering of all our coefficients:

	µ (k or s)
Experiment 1	0.400 (s)
Experiment 2	0.2769 (k)
Experiment 3	0.65 (s)
Experiment 4	0.154 (k)
Experiment 5	0.287 (k)

Some of the variations between the coefficients calculated from a flat and sloped surface could be due to the speed in the increment of the slope. We found that in several instances that the board would begin to slide down the sloped board at different angles. For experiments with hanging masses, we found that the block would move at different masses. There were instances where the block would begin to move when the hanging mass was up to 40 grams lighter than the case before.

Our calculated coefficient of kinetic friction in experiment 4 differed from experiments 2 and 5. This could be because the block was initially placed too close to our motion sensor.

Although there were many uncertainties in our experiments like the ones mentioned above, we were still accurately able to show that the coefficient for kinetic friction was larger than the coefficient for static friction.