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$$ \begin{array}{l}{\text { (II) Show that the following combination of the three funda- }} \\ {\text { mental constants of nature that we used in Example } 10 \text { of }} \\ {\text { "Introduction, Measurement, Estimating" (that is } G, c, \text { and } h} \\ {\text { forms a quantity with the dimensions of time: }}\end{array} $$$t_{\mathrm{P}}=\sqrt{\frac{G h}{c^{5}}}$This quantity, $t_{1}$ , is called the Planck time and is thought to be the carliest time, after the creation of the Universe, at which the currently known laws of physics can be applied.

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Physics 101 Mechanics

Chapter 1

Introduction, Measurement, Estimating

Physics Basics

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our question says show that the following combination of three fundamental constants of nature that we use an example 10 of introduction measurements estimating that is G, which is the gravitational constant C, which is thespian of light and H, which is place Constant form's quant, a quantity with the dimensions of time. Therefore, Tisa P is equal to the square root G over age divided by sea to the fifth this quantity Tisa P is called the planks time and is thought to be the earliest time after the creation of the universe at which the current known laws of physics can be applied. So I wrote here what we were given. I used brackets to represent the dimensions of the quantity. So if I put a bracket around the quantity, what I'm writing is equal to is the dimensions of that quantity. So the dimensions of capital G, which is a gravitational constant, is dimensions of length. Of the third divided by mass, which is capital M times, time squared the dimensions of H R mass times length squared, divided by time and the dimensions of the speed of light see our length divided by time. So then The question says show that T's A P has dimensions of time. Okay, so let's go ahead and do that. Let's find the dimensions of TCP. So write the racket because we're finding dimensions of TCP. Well, let's write what T's a P is equal to. I hope that's not a very dense, very simple what's right when it's equal to in terms of its dimensions. So the 1st 1 is G, which has dimensions of length cubed, divided by mass mass times time script. Okay, and then the dimensions of each core place constant our mass times lengths where extended version. Beirut divided by time. That's the numerator. And then in the denominator in the square, Root actually extends all the way down to the denominator. So let's go ahead and do that. Her good notation here want to race that line. Okay, so then, in the denominator white, the dimensions of sea to the fifth soc has length divided by time and all of this miss to the fifth. Okay, well, let's go ahead and simplify this now, while the masses air going to cancel the mass dimensions. Cancel. That's nice and easy length to Third Time's linked to the second gives us linked to the fifth, then t squared times T. Who's this? He doesn't care. Okay. And then all of that is divided by length over time to the fifth. So you have blanks divided by time, and then this whole thing is under the square root. Okay, so this can be re written again where the lengths cancel out, right? They're both to the fit. Now you're gonna have the square root of tea to the fifth, divided by T to the third. Okay? And that's equal to That's very true of T Square, which is t so therefore, we've just shown that the dimensions of TCP are indeed time.

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