# The smallest unit of length in physics is the Planck Length, denoted by ℓp. To determine the Planck Length, you need to know Planck's constant (h), the speed of light (c), and Newton's...

The smallest unit of length in physics is the Planck Length, denoted by ℓp. To determine the Planck Length, you need to know Planck's constant (h), the speed of light (c), and Newton's Gravitational Constant (G).

h=6.626*10^-34(kg*m^2/s)

G=6.674*10^-11(m^3/kg*s^2)

c=2.998*10(m/s)

ℓp=√h*G/2π*c^3

Find the Planck Length. Give your answer in scientific notation to 3 decimal places.

I'm completely lost with this problem, any help solving it would be greatly appreciated, thank you

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### 1 Answer

To determine the Planck length we use `lp=sqrt((hG)/(2pi c^3)) `

Here h is the Planck constant, approximately `6.626 "x" 10^(-34) ("kg""m"^2)/("s") `

G is the gravitational constant `6.674 "x"10^(-11) ("m"^3)/("kg""s"^2) `

c is the speed of light in a vacuum `2.998"x"10^8 "m"/"s" `

Note that the units are correct: `sqrt((("kg""m"^2)/"s"*("m"^3)/("kg""s"^2))/(("m"/"s")^3))="m" `

`hG~~6.626"x"10^(-34)*6.674"x"10^(-11)=44.221924"x"10^(-45) `

So `hG~~4.4221924"x"10^(-44) `

Also `2pi * c^3~~2(3.14159)*(2.998"x"10^8)^3=169.3067944"x"10^(24) `

So `2pi * c^3~~ 1.693067944 "x" 10^26 `

The radicand becomes `(4.4221924"x"10^(-44))/(1.693067944"x"10^26) `

This is approximately `2.611940304"x"10^(-70) `

Taking the square root of this yields:

`lp~~sqrt(2.611940304"x"10^(-70))~~1.61614984"x"10^(-35)"m" `

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`lp~~1.616"x"10^(-35)"m" `

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Note that when dealing with numbers in scientific notation (numbers of the form `a"x"10^b ` ) we can use the properties of real numbers -- specifically multiplication properties like commutivity. We can multiply the mantissas (the coefficients) and then multiply the powers -- to multiply powers with the same base you add exponents. Similarly, when dividing we divide the mantissas and divide the powers -- to divide powers with the same base we subtract the exponents.

Also, in the givens the value of c was incorrect. This might have been a typo, but if you tried to use this value for c you would not get the correct answer.

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