Show that the following combination of the three fundamental constants of nature that we used in Example 1–10 (that is G, c, and h) forms a quantity with the dimensions of time: tₚ = Gh\sqrt{Gh}/c⁵. This quantity, tₚ, is called the Planck time and is thought to be the earliest time, after the creation of the Universe, at which the currently known laws of physics can be applied.

Ch. 01 - Introduction, Measurement, Estimating

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

All textbooks

Giancoli Douglas 5th edition

Ch. 01 - Introduction, Measurement, Estimating

Problem 57a

Chapter 1, Problem 57a

Show that the following combination of the three fundamental constants of nature that we used in Example 1–10 (that is G, c, and h) forms a quantity with the dimensions of time: tₚ = $\sqrt{G h}$ /c⁵. This quantity, tₚ, is called the Planck time and is thought to be the earliest time, after the creation of the Universe, at which the currently known laws of physics can be applied.

Verified step by step guidance

Identify the three fundamental constants involved: G (gravitational constant), c (speed of light), and h (Planck's constant). Their respective dimensions are: G [L³M⁻¹T⁻²], c [LT⁻¹], and h [ML²T⁻¹].

The goal is to show that the combination tₚ = √(Gh/c⁵) has the dimensions of time. Start by writing the dimensional formula for the given expression: tₚ = √(Gh/c⁵).

Substitute the dimensional formulas of G, h, and c into the expression: G [L³M⁻¹T⁻²], h [ML²T⁻¹], and c [LT⁻¹]. The dimensional formula for tₚ becomes: tₚ = √([L³M⁻¹T⁻²] × [ML²T⁻¹] / [L⁵T⁻⁵]).

Simplify the dimensional expression step by step: Combine the terms in the numerator and denominator. The numerator becomes [L³M⁻¹T⁻²] × [ML²T⁻¹] = [L⁵M⁰T⁻³]. The denominator is [L⁵T⁻⁵]. Dividing these gives: [L⁵M⁰T⁻³] / [L⁵T⁻⁵] = [M⁰T²].

Take the square root of the simplified dimensional formula: √([M⁰T²]) = [T]. This shows that tₚ has the dimensions of time, as required.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Fundamental Constants

Fundamental constants are physical quantities that are universal in nature and do not change. In this context, G (gravitational constant), c (speed of light), and h (Planck's constant) are essential for understanding the relationships between different physical phenomena. These constants serve as the building blocks for various equations in physics, allowing for the derivation of other quantities, such as time.

Dimensional Analysis

Dimensional analysis is a mathematical technique used to convert one set of units to another and to check the consistency of equations. By analyzing the dimensions of physical quantities, one can determine if an equation is dimensionally correct. In this case, the analysis of the expression √(Gh/c⁵) helps to show that it has the dimensions of time, which is crucial for understanding the significance of the Planck time.

Planck Time

Planck time is a fundamental unit of time in the realm of quantum mechanics and cosmology, defined as the time it takes for light to travel one Planck length. It is approximately 5.39 × 10⁻⁴⁴ seconds and represents the earliest moment in the universe where classical physics breaks down, and quantum effects dominate. Understanding Planck time is essential for exploring the origins of the universe and the limits of our current physical theories.

Recommended video:

Guided course

05:59

Velocity-Time Graphs & Acceleration

Related Practice

Textbook Question

The speed v of an object is given by the equation v = At³ ― Bt, where t refers to time. What are the SI units for the constants A and B?

views

Textbook Question

American football uses a field that is 100.0 yd long, whereas a soccer field is 100.0 m long. Which field is longer, and by how much (give yards, meters, and percent)?

views

Textbook Question

Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary 'return force' for water piled up in the wave crests is due to the gravitational attraction of the Earth. Thus the speed v (m/s) of ocean waves depends on the acceleration due to gravity g. It is reasonable to expect that υ might also depend on water depth h and the wave's wavelength λ. Assume the wave speed is given by the functional form v = Cgᵅ hᵝ λᵞ, where α, β, $γ$ , and C are numbers without dimension. In shallow water, the speed of surface waves is found experimentally to be independent of the wavelength (i.e., γ = 0 in our assumed equation above for v). Using only dimensional analysis, determine the formula for the speed of waves in shallow water.

views

Textbook Question

One mole of atoms consists of 6.02 x 10²³ individual atoms. If a mole of atoms were spread uniformly over the Earth's surface, how many atoms would there be per square meter?

views

Textbook Question

Many sailboats are docked at a marina 4.4 km away on the opposite side of a lake. You stare at one of the sailboats because, when you are lying flat at the water's edge, you can just see its deck but none of the side of the sailboat. You then go to that sailboat on the other side of the lake and measure that the deck is 1.5 m above the level of the water. Using Fig. 1–14, where h = 1.5 m , estimate the radius R of the Earth.

views

Textbook Question

The following formula estimates an average person's lung capacity V (in liters, where 1 L = 10³ cm³): V = 4.1H ― 0.018A ―2.7, where H and A are the person's height (in meters) and age (in years), respectively. In this formula, what are the units of the numbers 4.1, 0.018, and 2.7?

views