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Simple Pendulum to lind '9' (4WWatahena] thic WYhde LonDtartur Problena fer Quz 45File CfUsers/bamir/Downloads/Pendulum pdfElttigu Llue to Dat Thc most co...

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Simple Pendulum to lind '9' (4WWatahena] thic WYhde LonDtartur Problena fer Quz 45File CfUsers/bamir/Downloads/Pendulum pdfElttigu Llue to Dat Thc most comnon Way to fit a line m + 6 mnnuzc thc following eulmdata points (*Y) (=ll_N itoS(m.b) - Zl;-y(x;))? J; = measured value y( x ) = fitted valueIt can bc show thatand harc gwvcn byEEy where A=NEs-(Esl' ECy_L Lv L=(lengt)(perio= )Type here L0 search

simple Pendulum to lind '9' (4 WWatah ena] thic WYhde Lon Dtartur Problena fer Quz 45 File CfUsers/bamir/Downloads/Pendulum pdf Elttigu Llue to Dat Thc most comnon Way to fit a line m + 6 mnnuzc thc following eulm data points (*Y) (=ll_N ito S(m.b) - Zl;-y(x;))? J; = measured value y( x ) = fitted value It can bc show that and harc gwvcn by EEy where A=NEs-(Esl' ECy_L Lv L= (lengt) (perio= ) Type here L0 search



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The swinging pendulum Refer to Exercise 33. Here is Minitab output from separate regression analyses of the two sets of transformed pendulum data:
Do each of the following for both transformations.
(a) Give the equation of the least-squares regression line. Define any variables you use.
(b) Use the model from part (a) to predict the period of a pendulum with length 80 centimeters. Show
your work.
(c) Interpret the value of s in context.

So we're trying to find a function for the angular displacement as a function of time. We can say that, um, we can, actually So this would be the system Ah, that we're working with. And so Ah, we can first say that the sum of the torque the net torque would be equal to the moment of inertia times the angular acceleration. This would be equal to minus mgl sign of fada. And this would be equal to ay the moment of inertia times the second derivative of theano angular displacement with respect to time. So that's simply just the, ah definition of the angular acceleration. And if the angular displacement is limited to 15 degrees So if this right here, if they do, Max simply equals 15 degrees. Ah, we can make the small angle approximation and say that sign of Fada is approximately equal to theta. Therefore, minus mgl Fada would be equal to I d. Listen, the moment of inertia times a second derivative of Leda with respect to time s so we can say D, the second derivative of Fada with respect to time would be equal to negative. M g l over I times Fada. Let's get a new brick book here and say that the second derivative will then be equal to negative M g l over. And I'll square the moment of inertia Times data, and this is just gonna equal negative G fada over. L uh, We can notice that this is ah, modeling simple harmonic motion where omega squared equals G over l. So given that, given that this model simple harmonic motion, we can use equation 14 4 and say that theta equals fate A Max Times Co sign of Omega T plus five, the face constant. And so Fada will equal fate of Max Times Co sign of t times the square root of G over l plus the face constant. And so this would be our final answer. That is the end of the solution. Thank you for watching.

So we're going to use the same approach. Taken in part B of sample problem 15 5 Uh, this is the physical pendulum period and length problem, but we're going to treat ah, in our case, a more general possibility for the moment of inertia. So essentially we can say that t the period would be equal to two pi times the square root of if the length divided by G and this is gonna be equal to two pi times the square root of the moment of inertia. I divided by m g. H. And so this would be equal to rather this we need to actually, um, solve for Oh, not so we can eliminate the two pi. And we're going to see that after canceling out the squaring both sides, we find that how not over G would be equal to ay over mg h. Of course, the gravity cancels out and we have that the length would be equal to the moment of inertia over m h. This would be our proof. That is the end of the solution. Thank you for watching

Hello and welcome to this video solution of numerous. Here we are given a physical pendulum that as a center of mars So let's say this is a physical pendulum that has a center of Operational Distance to Elway three from its point of suspension. So let's say the point of suspension is this and we have caught the center of oscillation. Let's say this is a thing. So this is a distance This has given us to well by three. Now you have to show that the distance between the point of suspension and the center of oscillation for a physical pendulum of any form is able. It's the formidable image is the moment of furniture, no red and it is the distance between the pivot and dependent on center plus. So this is a judge, right? Uh huh. Now here we have to calculate its form rate of the lamp so we can have it the total time period equal. Do bye go to hell with the right is the expression of simple pendulum. Now, for a compound where the same equation is to Beirut overall, are you breaking teach comparing both the equation? It is more evident that talent L. Equal to I buy average right? This L is equal is the distance between the point of suspension of the center of oscillation. For the physical pendulum. I hope this is clear to you and have a very good rest of the day. Thank you.

From this problem. We're doing a simple pencil. I think you know about a simple pendulum is that its motion is simple harmonic motion meaning that also Leight's or it moves backing fourth, and it's at a certain link. And so we also want to know it's period or the time it takes for each oscillation that happen. So just from that, you know that you find a period, we just need to know the time and the number of oscillations in that time period. Let's look up what we're giving. So we're given three different links and with each different with three different times. So we have one 1 m and then here we have t one 99 0.8 seconds. Next we have l two, that's 0.75 meters, the Hear t two IHS 86.6 seconds and then finally l three 0.5 m near a T three is 71 0.1 seconds. They were also till the breaks beach time. There were 50 oscillations. So for part A, we can use that equation because you want to find the period at each length and so were given the time and we also know for each one it took 50 oscillations. So t one, we have 99.8, the better by 50. So here we get two seconds. Next T two 86.6 divided by 30. So here it's 1.7 seconds and then t three 71.1 divided by 50. So t three, it is 1.42 seconds. Okay, it's now for part B. You want to calculate gravity for each length and period on the way to do that is to use another equation to find the period which has given us t is equal to to buy square root of they'll divided by G So square both sides get into the square room. We have t squared is equal to floor. Hi square. That's allergy. So then selling for gravity we have she is equal to four pi squared time now divided by D squared. Okay, so for G one at length one and t one, it's for pie square times one divided by two squared. So G one 9.87 music per second squared next G to so again l two and t two. It's for pie squared times 75 divided by 1.7. So a G two is 9.89 meters. Your second square. Okay, enough of G three again. Lcv end t three. It's foreplay squared this 0.5 divided by one point for two square. So RG three is 9.79 meters per second squared. No, we want to do now is get the mean value off these gravity's we just calculated and compare it to what we know, which is. Gravity is equal to 9.8. So let's call this g mean so to do this, we just add up g one plus g two g three and divide them by three. So we have 9.82 What for? 89 plus 9.79 The writer by three So g mean is nine point 85 meters per second square. So doing a percent ever with what we know. So it's 9.8 minus 9 25 this is the opposite value their water by name. Honey. Times 100. So our percent error, it's by percent. Okay, now we're see Want to plot a T square versus l graph? and then from that ground You want to calculate gravity and again compare it What we know, which is 9.8 so t squared. These are y values. L is our X value. So l we have quick by 0.75 and we have one. And now for T Squares, we have 2.1 2.89 and then we have four because we took the periods that we have And then did we square them? So if you look at these values, you can see that l R value is increasing constantly and t squared over y Value is increasing constantly as well. So that means our graph she looks something like this. They should have a constant slope, you know, to find G. You look back at the equation that we found for tea. And again if we squared both sides to get rid of the square room, you have t squared more pie squared. L f G c Just move the G over there under the four pi squared. So you have t squared is equal to four by squared over G. So looking at this equation, we see okay, He squares why value that's equal to over here. L is our X value so than this must be our end. Wise equals MX. So that means we know that the slope is equal to for pie squared over g solving for the slope or something for G m e. We get g is equal to were quite square divided by slope. So to find the slow we take any number, any two numbers are in two sets of numbers from the graph we have because the slope is equal to y tu minus Phi one A lot of my ex two minus x one. So then we can choose any of them. So let's say it for minus 2.89 divided by one minus 10.7 fire. So our slope is 4.44 So then working that in to our equation for Gene, we have jeez equal to four ply squares. We were for four for so here G is equal to 9.0 meters per second squares. So then comparing that to what we know, being a percent ever, we have mind pointing minus put o absolute value divided by 9.8 times 100. So our percent error here is eight percent


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