Solve each problem. Period of a PendulumThe period of a pendulum ...

Question

Solve each problem. Period of a PendulumThe period of a pendulum varies directly as the square rootof the length of the pendulum and inversely as the square root of the accelerationdue to gravity. Find the period when the length is 121 cm and the acceleration due to gravity is 980 cm per second squared, if the period is 6π seconds when the length is 289 cm and the acceleration due to gravity is 980 cm per second squared.

Accepted Answer

Hey, everyone in this problem, Summit is performing a physics experiment where he's trying to find the time period of a pendulum. OK. He knows that that period is directly proportional to the square root of the length and inversely proportional to the square root of the acceleration due to gravity. OK. So let's stop there and write out what we've been told so far and we're gonna come back to the question and figure out what it's asking. But what we're told is that the period is directly proportional to the square root of L. OK. The length, the square root of the length which you're calling L and it's inversely proportional to the square root of the acceleration due to gravity. If it's inversely proportional, that means that it's proportional to one divided by that value. OK. So our period is gonna be proportional to the square root of L divided by the square root of the acceleration due to gravity, which we're gonna call G. All right. So getting back to our question, we're told that this value of G is a constant that's equal to 9.8 m per second squared. Now summit's only gonna change the length of the thread for that pendulum. He found that the time period for the pendulum is four pi seconds when he uses one m of thread. And we have to find the period of the pendulum. If he uses 1.5 m of threat, we're given four answer choices. Option A eight pi multiplied by the square root of 1.5. Option B four pi divided by the square root of 1.5. Option C four pi multiplied by the square root of 1.5. And option D four multiplied by the square root of 1.5. And all of those answer choices are in seconds. Now, we've written our proportionality here. What we want to do is convert this into an equation so that we can use it to solve for the period. In order to convert this into an equation, we need to introduce a proportionality constant. OK. We can replace our proportionality symbol with an equal sign. So we can write that the period is equal to OK, the square root of L divided by the square root of G. But we have to multiply that by a constant C in that proportionality constant. Now we're looking for the period for a particular length L. So we know L we know the acceleration due to gravity G but in order to calculate the period, we still need to know what C is. So we're first gonna have to use the other piece of information we're given and we're given a particular period for a particular string light. We're gonna use that to figure out this value of C. Then we can use that to calculate our period. All right. So again, the period which we're given is 45 seconds. And that is for a length L of one m. So let's substitute this into our equation and see what C will be. We get four pi seconds. It's going to be equal to that constancy multiplied by the square root of one m divided by the square root of 9.8 m per second squared. We want a sulfur C. So we're gonna multiply by the square root of 9.8 m per second squared. We're gonna divide by the square root of one m. We're gonna have that C is equal to four pi multiplied by the square root of 9.8. And you'll notice that I don't have any units here. And that's because those units are gonna cancel out, ok. On the right hand side right now, we have the square root of a meter divided by square root of meter per second square. OK? So that square root of meter is gonna divide out. Then we get the square root of per second squared. That's gonna give us a unit of per second. We multiply that to our unit of seconds that we have for a period. Those are gonna divide it. OK. So the value of C is actually gonna be unit list. OK? All of those units have divided it and one left ac is equal to four pi multiplied by the square root of 9.8. Now we're gonna leave it like that. We're not gonna round it just so they can avoid round off error. And we're gonna move over to calculating the period we really wanted to find. So that period, OK? It's gonna be equal to this value of C which is four pi multiplied by this square root of 9. multiplied by the square root of L divided by the square root of G. Now, in this case, we want to find the period for a length of 1.5 m. So we get four pi multiplied by the square root of 9. multiplied by the square root of 1.5 m all divided by the square root of 9. m per second squared. Now, this square root of 9.8 that's gonna divide up the unit of square root of meter is going to divide out. We're gonna be left with the unit of per second squared, OK? The square root of per second squared in the denominator, we take the square root and we get a unit of per second because we're dividing by that, OK? We can multiply by the reciprocal. So we're gonna get a unit of seconds in the numerator, which is exactly what we want. And what we're left with here is four pi multiplied by the square root of 1.5 seconds. And that is the period we were looking for for a string length of 1. m. If we compare this to our answer choices, we can see that this matches with option C Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Direct Variation

Inverse Variation

Combining Variations

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