Kater’s Pendulum
Thomas Markovich and Kapil Chhabria
Departments of Physics University of Houston Houston, TX 77204-5006 (Dated: December 9, 2010)
We experimentally determined the local gravitational constant using Kater’s Pendulum to provide the authors with experience in data analysis. In this manuscript, we rigorously derive the relevant equations from first principles with the appropriate expressions for the experimental uncertainty. We found that by assuming the periods were equal we were able to determine that g was 9.7993 ± 0.0010 m/s2 which was 0.0653 % from the known value. If we took into account the period difference, we found that g was 9.7982 ± 0.0083 m/s2 which is 0.0552 %. Both methods gave us results well within our
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If we first begin by recognizing that Ic = mk 2 where k is the radius of gyration, we get that T1 = 2π and T2 = 2π
2 k 2 + l2 . dg 2 k 2 + l1 dg
(II.15)
(II.16)
(II.17)
FIG. 1. Kater’s pendulum. I1 is the moment of inertia about K1 with l1 is the distance from the knife edge to the small weight and I2 is the moment of inertia about K2 where w1 > w2
If we simply square our periods and solve for our radius of gyration in both equations, we find
2 4π 2 T 2 l1 − T2 l2 = 1 2 . 2 g l1 − l2
(II.18)
If we invert the system, we find that T1 = 2π and T2 = 2π
2 I + M I2 . M dg 2 Ic + M l1 M dg
(II.8)
2 g l2 − l2 = 21 2 . 2 4π T1 l1 − T2 l2
(II.19)
This is a familiar algebraic form and we solve it by separating our equation into (II.9)
2 2 l1 − l2 A B 2l − T 2l = L + l − l . T1 1 1 2 2 2
(II.20)
If we adjust the weights until the periods are equal, we simply get that
2 2 Ic + M l1 I + M I2 = M dg M dg
2 And after some algebra, we get that A+B = T1 and B2 A=−T2 and yields the following equation for g.
(II.10)
g=
8π 2
2 2 T1 +T2 L
+
2 2 T1 −T2 2l1 −L
(II.21)
3 This equation has an associated uncertainty of (∂T1 gδT1 ) + (∂T2 gδT2 ) + (∂L gδL ) + (∂l1 gδl1 ) (II.22) or δg = (
2 2 256π 4 −T2 + T1 2 2 δl
2 2 T2 +T1 L
2
2
2
2
(2l − L)4
2 2 −T2 +T1 2l−L
4
+ +
64π 4 +
2T1
For the reaction_, determine how many moles of chlorine Cl2 would be needed to react with
The parcel starts descending at 6000’ feet, the temperature at this height is 50.1 F, and is above the original LCL. The new LCL is at 6000’ feet as the parcel is starting its descent down the lee side of the mountain. The mixing ration to start to movement up the mountain was approx 18.9 g/kg and the descent capacity is approx 7.6 g/kg.
l19 – 18.33l = .67, l17 – 18.33l = 1.33, l19 – 18.33l = .67,
P B - 2 0 0 6 - 2 | M a y 17, 2 0 0 6
The temporal setting “oppress the character with the shape of a pendulum” (3) He fears its deadly velocity which represents his final hours of life. He feels terror of the doom that will “cut” his time on earth. As everyone knows, this symbolizes that death is inevitable.
Enter the precise mass in GRAMS of the potassium iodate used to prepare your primary standard solution.
S2M = ( M-GM) 2 = (5-5)2 +(4-5)2 + (6-5)2 = 0+1+1 = 2 = 1
The purpose of this experiment was to create a model that showed the motion of a spinning body and to test the accuracy of said model. The model was created by calculating the measurements of the orbital period of a stopping mass for six varying mass ratios and then comparing them to the model equation derived from the equation for centripetal force. The primary results of the lab were in agreement with the predicted values, therefore our degree of confidence in our data was high.
molality m= (grams MgCl2÷95.21gmol of MgCl2)10mL H2O×1g H2O1 mL H2O×1 kg H2O1000 g H2O+grams of ice×% of ice melted×1 kg ice1000 g ice
Still I was a bit lost, I went on and to have another reading to what was I supposed to do in the assignment, I decide to use the example of finding the circle area, a copied and pasted, but it did not work, I only got a syntax error, I keep playing with it and decide to use the forum
I found the secret formula, it was (w+L)-2 but w/l had to be reduced so it
. The second term in first equations ([Ch2E-15]) and ([Ch2E-16]) becomes much smaller than the first term, because the detuning \Delta=\omega-\omega_{\textrm{A}}\ll\omega_{\textrm{A}}
From the equation, we know that 2 moles of HCl reacts with every mole of Mg.
Introduction During this lab you will become more familiar with the concepts of torque. The purpose of this lab is to determine if the rotational equilibrium condition, Στ = 0, holds experimentally. Equipment Meter stick (1) - no metal ends Fulcrum (1) Clamps (4) Weight Hanger (1) Mass Set (1) Digital Scale (1)
A simple pendulum consists of a mass that is attached to a string of length ‘L’ that is fixed to a point, in this case, a cork suspended by a clamp stand. This allows the mass to be suspended vertically downwards and allows it to be displayed at an angle that it swings. A period ‘T’ of oscillation is the time required for one complete swing. For this to happen ideally its mass must swing from an angle that is