00:01
Hi there, so for this problem we are told that the period of a simple pendulum of length capital l is given by the following.
00:09
So the period is 2 times pi times the square root of the length divided by the acceleration due to gravity, the square root of that.
00:19
We assume that the acceleration due to gravity on the surface of the earth, it is a constant value of 32 feet per second squared.
00:32
If the pendulum is that of a clock that keeps good time and that its initial length is 4 feet, and how much time will the clock gain in 24 hours? if the length of the pendulum is decreased to a value of 3 .97.
01:04
So then in here we can already obtain that delta l is just simply the difference between the final value that is 3 .97.
01:36
Oh sorry this was l sub 0 and then this l will be 3 .97 minus 4 so this will be um 3 .97 minus 4 so this will give us minus 0 .03 so that will be the decrease and the length for for this.
02:01
Once we have this, we need to determine how much time will the cloud gain in 24 hours.
02:12
First, let's determine how is the period of this.
02:14
The period is 2 times pi, the square root of the length, initial length that is 4, divided by the acceleration due to gravity that is 32.
02:23
So then in here, simplifying this, we will obtain that the period for this is equal to 2 .22 this will be in units of well let me just put it this exactly as just pi divided by the square root of 2 in seconds okay so then now to find how how much this has changed.
03:00
So the change in the period is the absolute value of the period with respect to time.
03:09
Well, with respect to, sorry, with respect to the length in here, times the change in the length.
03:15
So the partial derivative of this with respect to the length is, we can treat the square root of the acceleration due to gravity as a constant.
03:24
So we will have the derivative of this square root of the length.
03:30
So that will give us just simply 1 divided by 2.
03:35
That's cancelled that 2 in here.
03:40
And then this times 1 divided by this square root of the length.
03:45
So we will have just simply this in here.
03:56
And that's it...