8 An experiment to determine atmospheric pressure P uses the...

8 An experiment to determine atmospheric pressure P uses the equation P = pgh where p = (13 800 ± 200) kg m$^{-3}$, g = (9.81 ± 0.03) m s$^{-2}$, h = (0.788 ± 0.006) m What is the value of P, with its uncertainty, when stated to an appropriate number of significant figures? A (1.0168 ± 0.2685) × 10$^5$ Pa B (1.067 ± 0.025) × 10$^5$ Pa C (1.07 ± 2.5%) × 10$^5$ Pa D (1.1 ± 0.3) × 10$^5$ Pa

Transcript

00:01 Hi there, so for this problem we are given an experiment with the following information.

00:05 The density is 13 ,200 plus or minus 200.

00:10 This 200 is the uncertainty.

00:13 This is in kilograms per cubic meter.

00:15 The acceleration due to gravity is 9 .81.

00:20 This plus or minus 0 .03.

00:23 This in units of meters per second squared.

00:25 Square and the height that we are also given 0 .788 plus or minus 0 .006 in units of meters.

00:35 So the question is what is the value of p with its own surrogacy when it's dated to an appropriate number of significant figures? now remember that well we already know that the pressure is the pressure between the density, the acceleration due to gravity and the height.

00:50 You get that right.

00:52 So for example to find this we just do the product between these three values so that will be 13 ,200 this times 9 .81 and then this times 0 .788 so when we do this product it, that will be 13 ,200 times 9 .81 times 0 .788, so that will give us 1 ,000, oh sorry, i made a i'm mistaken here is 13 ,800.

01:36 Okay.

01:36 So let me fix that.

01:38 That will give us 106 ,677 .86.

01:50 Okay.

01:51 We can write this as 1 .07.

01:55 Now we write it like this because we want it with the least decimal places.

02:01 And the one that has the least decimal places is acceleration due to gravity that has three decimal places.

02:07 So we need to put that as 1 .07 times 10 to the 5.

02:13 Okay? hi.

02:15 And now we have done this.

02:16 We need to calculate the uncertainty.

02:19 Now, remember that the uncertainty is not calculated by adding the percentage together.

02:26 We need to do the following.

02:29 We need to do the partial derivative of this in magnitude with respect to one of the quantities, for example, with the density.

02:41 And then this times the uncertainty of the density plus the partial derivative of this with respect to the acceleration due to gravity, the uncertainty with the acceleration due to gravity, and the same with the height.

02:57 Now, these are, well, in this case, with the expression that we are given, these are quite easy to obtain.

03:04 The partial derivative with respect to the density is just, we just differentiate this expression with respect to the density, leaving the address comes to, so that will give us acceleration due to gravity times the height...

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