Theory: Because the apparatus is made of three cylinders, and we know the total mass of the apparatus, the moment of inertia can be determined with careful measurements of the length and radius of the three cylinders to determine the volume(V=Length*Pi*Radius^2) of each individual cylinder. We can determine the mass of each cylinder with this information, and therefore calculate the total moment of inertia (the moment of inertia for a cylinder is =0.5*Mass*Radius^2) by adding the individual moments of inertia of each cylinder. The angular acceleration can be determined by finding the angular velocity (omega=W), which can be found by determining the velocity of the apparatus as it spins. With angular acceleration and the moment of inertia we can determine the torque on the apparatus (Torque=Inertia*Angular acceleration), due to the force of the cart pulling on the apparatus as it descends (there will be friction), with these now known forces (using force diagrams) determine the acceleration of the cart, and use the kinematics equation (Distance=0.5*acceleration*time^2) to calculate the time it would take for the cart to travel down the sloped track with less than 4% error.
Procedure: Determine the volume of the apparatus by taking measurements, realize the percentage of volume will correlate with percentage of mass of each cylinder of the apparatus. Calculate the moment of inertia for the entire apparatus as I_total= I_1+I_2 +I_3. Use logger pro to determine the angular acceleration of the apparatus as it slows. Slant a 1 meter track against the table, place a 503 gram cart at the top of the track, and tie a string from the cart around a cylinder on the apparatus. Time how long it takes for the cart to descend 1 meter and compare the result with the calculated time that your measurements suggest the time would be.
Apparatus
Measurements of Apparatus
*Total mass 4808 grams
*Cylinder #1: Length=5.0 +/- 0.01 cm, Radius= 1.55+/- 0.01 cm, Volume= 37.7 cm^3
*Cylinder #2: Length=5.15 +/- 0.01 cm, Radius= 1.55+/- 0.01 cm, Volume= 38.9 cm^3
*Cylinder #3: Length=1.55 +/- 0.01 cm, Radius= 10.025+/- 0.01 cm, Volume= 489.38 cm^3
* Total Volume 565.98 cm^3
* Percentage of Volume of each cylinder to the total should be the equivalent of percentage of mass.
Volume percentage= Volume individual/Volume total*100%
Mass individual= (Volume percentage/100) *Total mass
*Cylinder #1 percentage of volume 6.66%, so mass equals 320.3 grams
*Cylinder #2 percentage of volume 6.87%, so mass equals 330.4 grams
*Cylinder #3 percentage of volume 86.5%, so mass equals 4157.3 grams
Caption of Apparatus as it decelerates using logger pro
Graph of X,Y Velocity and Omega vs time
*Omega can be calculated as = sqr(Velocity_x^2 +Velocity_y^2)
*Once graphed as Omega vs time, the slope will give us angular acceleration (a_a)
*Angular acceleration (a_a) is -0.4063 m/s^2
The Track and Force Diagrams
*Angle track makes with the horizontal is 47.5 degrees
*Force diagram on track
*Actual time it takes for cart to reach end of 1 meter track= 7.16 seconds.
Calculations
*Total of moment of inertia
*Calculating time
*Calculated time was 6.98 seconds
*Difference from actual (Actual time-Calculated time/Actual time *100%) was 2.93%
Conclusion
*Since we fell within the necessary 4% error margin, I say the experiment confirmed my methods of determining the moment of inertia of the apparatus and its angular acceleration.
*I used difference percentage, instead of propagated error, perhaps my result would've have been more accurate had I used this method, but I was satisfied with the 2.93% of the difference.
*Although the calculations took frictional torque into account, it did not take friction from air resistance into account or friction from the track.
No comments:
Post a Comment