Wednesday, December 7, 2016

Lab# 22: Physical Pendulum Lab

Physical Pendulum 

Samuel
Ellis
Mia
11-28-2016

Purpose:

Derived expressions for the period of various physical pendulum. Verify your predicted period by experiment.

Theory:


Procedure:

1. derive expressions for the moment of inertia of the ring about this point and the period of this ring behaving as a physical pendulum
2. Make physical measurement of the relevant quantity for the ring
3. Measure the actual period of the ring, acting as a physical pendulum
4. compare your experimental result with your theoretical result
5. Cut our or find objects of the appropriate shapes. Measure the appropriate dimensions of each one, and their respective masses
6. Attach a thin piece of making tape to the bottom of each object
7. Use the same setup as above to measure the small-amplitude period of oscillation for each one.
8. Compare the actual and theoretical values for the periods using the actual measured dimensions for the various object.


Measured Data:

Ring_inner = 20.59 cm
Ring_outer = 21.8 cm
Height_1 = 12 cm
Based_1 = 13 cm
Height_2 = 13 cm
Based_2 = 13.4 cm

Graph:

 (Above, is the oscillating ring graph. The period slowly decreasing due to frictional force acting between rod and and the ring.) 

 (Above, is upright triangle graph. This graph has lesser decreased because we used paper clip instead of metal ring. So the frictional force is lesser.)

(Above, is upside down triangle. The period is faster than the upright triangle. Because it has less mass on the bottom)


Calculation:

I_pt = I_cm + m*r^2
        = m*r^2 + m*r^2
        = 2*m*r^2

I = I*๐ฐ
m*g*sin๐›‰*r = 2*m*r^2*๐ฐ
๐ฐ = g/2*r * sin๐›‰
w^2 = g/(2*r)
w = sqrt(g/(2*r))

Based on above equation, I got 0.928 for my calculated period for the ring. 

Conclusion:

In this experiment, we derived a equation to determined the period of oscillating ring and compared it with actual period of the ring. As usual, we have made some assumption. First, the angle we pushed can not be larger than 10 degree. Because if it is less than 10 degree, the amount of error is so small, we can just ignore it. Secondly, we assumed there's no frictional force acting between two materials. And lastly, we assumed the rod for holding the object is parallel to the surface. 



      

Monday, December 5, 2016

Lab#21: Mass-Spring Oscillations

Mass-Spring Oscillation

Samuel
Ellis
Mia
11-23-2016

Purpose:

To determine the oscillation of the spring with different weight and compare it to see if simple harmonic equation is accurate.

Theory:

Based on simple harmonic equation, we know that spring and pendulum have same oscillation. And we know that oscillation is based on how much the spring stretched. When releasing spring and weight dropped, the up and down motion is determined by the spring constant. Since we have different spring with different spring constant, we can use oscillation equation to predict it period. 

Procedure:

1. Mount clap on the table, with a rod horizontal respect to the table. 
2. Hang a spring on the rod.
3. Hang 5 different weigh on it and record it period and mass.
4. Find spring constant of the spring.


Measured Data:

Data for Spring 4
m = 215 g
t_1 = 6.66 sec, t_2 = 6.63 sec, t_3 = 6.62   t_average = 6.64 sec

m = 315 g
t_1 = 8.05 sec, t_2 = 8.00 sec, t_3 = 8.08 sec  t_average = 8.05

m = 4.15 g
t_1 = 9.23 sec, t_2 = 9.21 sec, t_3 = 9.21 sec  t_average = 9.21 sec

m = 65 g
t_1 = 3.63 sec, t_2 = 3.62 sec, t_3 = 3.65 sec  t_average = 3.63 sec

Spring for All Group

Calculation:

Predicted value:
T = 2*pi*sqrt(m/k)
T = 2*3.14*sqrt(0.115/19.4)
T = 0.48 sec

Measured value:
T = time/how many times
T = 5.22/10
T = 0.522 sec

Analysis:

1. We calculate our spring by using spring potential energy equation, 0.5*k*x^2. k is spring constant and x is how much it stretch. 

2. 

3.  

5. Our spring constant is right. But if k is off by 5% or more, my first option will go with period. Usually we measure period for at least 3 times and those time should somewhere similar to each other. We then average those number to get most accurate data. However, when averaging 3 different number, there's will be uncertainty, and that's where those error come from.

6. According to our formula, period(T) = 2*pi*sqrt(m/k). In this case, m is mass & k is spring constant. k = (2*pi*m)/T. since 2*pi is constant, we can simply ignore it when determining variables. Since k = m/T(when ignoring 2*pi), when T is decreasing, it will make m divide by smaller number, which will have bigger product. In this case, it will be k. 

7. Use the same equation from above, k = m/T. We know that if we increase the mass(m), then k will become bigger number. Since m is increasing, bigger number divide by smaller number result in bigger resultant value. Therefore, when increasing period, mass is also increase.

Conclusion:

In this experiment, we used simple harmonic equation to find the period and spring constant of the spring. We used different weight and make it oscillating so we make sure our spring constant is the same. Thus, we used the same equation to calculate it period. However, the number we got are little bit different from the actual measured data. 0.48(calculated value) verses 0.522(measured value). And that difference might due to the rounding error of the measured period or the human error when stopping the timer. 




Lab#20: Conservation of Linear and Angular Momentum

Conservation of Linear and Angular Momentum

Samuel
Ellis
Mia
11-21-2016

Purpose:

To understand how conservation of momentum can translate to both linear and angular direction.

Theory: 

When object is rotating, it will have a tangential and linear direction. When the tension is release, the object will move toward tangential direction. And since we know that momentum is conserved, therefore, we can use tangential momentum and change that to angular momentum. In addition, we can use kinematic equation to find how far or how long object travel. Because velocity is the same throughout flying period.  


Procedure:

1. Use the aluminum to disk, and mount the ball-catcher on top of the small torque pulley using a gray-capped thumbscrew.
2. Using the method of Experiment 6, determine the moment of inertia of the disk and ball catcher. Also measure of the ball. 
3. Place the ramp on the edge of a table. Determine the horizontal velocity of the ball as it roll off the end of the ramp
4. Mark a starting point on the ramp
5. Release the ball from this staring point and note where the ball strikes the floor(a floor of carbon paper over a sheet of white paper will leave a mark for measuring)
6. Measure the distance L and h. The horizontal velocity of the ball is then equal to L/t where t = 2*h/g.
7. If you have time, you may want to calibrate the ramp by determining the horizontal velocity for a variety of starting points. If you plan on doing the optional part of the analysis for this experiment, you will also need to record the starting height of the ball. 
8. Roll the ball down the ramp so that it is caught by the ball-catcher. Quickly remove the ramp, so it doesn't disturb the rotational of the ball-catcher, and record the reading of the digital display. Record the horizontal velocity of the ball based on your earlier measurement, and also record the radius.
9. Repeat your measurement.
10. Using procedure outlined above, make the measurement necessary to investigate the relationship between the angular momentum imparted to the system.


Measured Data:

Mass_ball = 28.9 grams
Diameter_ball = 19 mm
h = 95 cm
L = 59.5 cm
d = 16 cm
hang mass = 24.6 g
diameter_pulley = 47.9 mm
w = 2.269 rad/s
r_ball to axis = 7.6 cm

Graph:

(Above, is the graph of ball-catcher rotating on top of disk. The consistent slope is cause by the rotating disk. Since it's angular velocity is the similar, so the slope looks almost identical.)


(Above, is the angular acceleration graph of it going up and down.)

Calculated Data:

Find hanging time in air
ฮ”y = -0.5*g*t^2
0.95 = -4.9*t^2
t = sqrt(0.95/4.9)
t = 0.440

ฮ”x = v*t
0.59*t = v*(0.440)
v = 1.35 m/s

Find w
1.
(0.076)(0.02898)(1.35) = (1.05*10^-3)+0.0289*0.076^2+ 2/5 *0.0289*0.009^2
w = (0.002965/0.001217)
w = 2.43 rad/s

2.
0.042*0.0289*1.35 = 0.00105 + 0.0289*0.042^2 + 2/5 *0.0289 * 0.009^2
w = (0.001638/0.001102)
w = 1.48 rad/s

Conclusion:

In this experiment, we used already prepared equipment to conduct this experiment. We used LoggerPro to calculated the angular velocity of the rotating disk with ball-catcher. We then measured the height, distance and length of the ramp. With all the measured data, we are able to use kinematic equation to find the air time during the fall. And according to conservation of momentum, all energy conserved. So for our calculated data for angular momentum, we got 2.43 and 1.48. And for theoretical we got 2.43 and 1.49. Which is almost identical and it also prove momentum is conserved.