Still learning dimensional analysis

[help1please]

Homework Statement

angular frequency omega, speed of light and gravitational constant

Homework Equations

Just the expression,

$$\frac{2 \omega c}{\sqrt{G}}$$

The Attempt at a Solution

Still learning dimensional analysis. So I am simply wanting to know if I have done this right. If I have, I get

$$\frac{2 \omega c}{\sqrt{G}}$$

this with dimensions of force, right?

Thanks in advance.

 

[cepheid]

Homework Statement

angular frequency omega, speed of light and gravitational constant

Homework Equations

Just the expression,

$$\frac{2 \omega c}{\sqrt{G}}$$

The Attempt at a Solution

Still learning dimensional analysis. So I am simply wanting to know if I have done this right. If I have, I get

$$\frac{2 \omega c}{\sqrt{G}}$$

this with dimensions of force, right?

Thanks in advance.

I'll use square brackets around a quantity to mean, "dimensions of" that quantity (which is a fairly common notation, I think). Then:

[ω] = time^-1

[c] = length * time^-1

[√G] = [G]^1/2 = (force * length² * mass^-2)^1/2

Therefore [ωcG^-1/2] = length * time^-2 * force^-1/2 * length^-1 * mass

= force^-1/2 * mass * time^-2

= (mass * length * time^-2)^-1/2 * mass * time^-2

= mass^1/2 * length^-1/2 * time^-1

So, no, it doesn't have dimensions of force.

 

[help1please]

What does it have dimensions of then? I mean, other than what you have said, is there a commonly known dimension it exhibits? like energy for instance?

 

[cepheid]

It doesn't correspond to any common physical quantity, and there is no reason that it has to (you've simply come across a new type of quantity),

However, the *square* of the quantity is a common type of quantity. Hint: square everything in blue, and rearrange things to produce "force" plus some leftover stuff. What is the dimension of the quantity you've come up with?

 

[Nickie]

I would like to commend you for taking the time to understand and learn dimensional analysis. It is a crucial tool for any scientist and will greatly benefit your understanding of physical equations.

In regards to the expression \frac{2 \omega c}{\sqrt{G}}, you are correct in saying that it has dimensions of force. To be more specific, it has units of Newtons (N) since the units of angular frequency (rad/s) and the speed of light (m/s) cancel out the units of the gravitational constant (m^3/kg/s^2). This expression is often used in the context of gravitational force, where the force is equal to the product of the mass of an object and the acceleration due to gravity, which can be expressed as \frac{2 \omega c}{\sqrt{G}}.

I hope this helps and keep up the good work in learning dimensional analysis!