What must the Dimensions of G be for this equation to be dimensionally correct?

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In summary, an object with escape speed ve from the Earth can move infinitely far, regardless of its mass, as long as it depends on G, the gravitational constant, ME, the mass of the Earth, and RE, the radius of the Earth. The dimensions of G must be such that the equation is dimensionally correct. For the equation for T, the period of a planet around the Sun to be dimensionally correct, the units of G must be determined, as well as the values of a and b for the equation. The problem is asking for a lesson on how to solve this, as the textbook and lessons given do not provide sufficient explanation.
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Homework Statement



"An object from the surface of the Earth with the escape speed ve can move infinitely far from the Earth. The escape speed is independent of the mass that is trying to escape and only depends on G, the gravitational constant, ME the mass of the Earth, and RE the radius of the Earth.

a.) What must be the dimensions of G be for this equation to be dimensionally correct?

b.) Given your units for G in part (a), what must the values of a and b be for the equation for T, the period of a planet around the Sun, below to be dimensionally correct?
T = 2*π*G^(-1/2)*M^a*r^b

Here M is the mass of the Sun and r the distance the planet is from the sun. The power of G is already given at -1/2.

Homework Equations



unknown

The Attempt at a Solution



None. This "class" consists of a "textbook" that contains only physics problems and no explanations as to how to do them, and "lessons" that consist only of the teacher writing problems on a white board and us having to solve them in class. I need a full lesson on how to do this, but am clueless as to where to find it, or even what to look for. I'm not even sure what the question is asking.
 
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What's the first equation the problem is referring to?
 

FAQ: What must the Dimensions of G be for this equation to be dimensionally correct?

What is the purpose of considering the dimensions of G in an equation?

The dimensions of G, also known as the gravitational constant, play a crucial role in determining the accuracy and validity of an equation. By ensuring that the dimensions of G are correct, we can ensure that the equation is consistent and physically meaningful.

How do you determine the dimensions of G?

The dimensions of G can be determined by analyzing the units of the equation in which it appears. By using the principle of dimensional analysis, we can break down the equation into its fundamental units (such as length, mass, and time) and determine the dimensions of G that would make the equation dimensionally consistent.

Can the dimensions of G change depending on the equation it is used in?

Yes, the dimensions of G can vary depending on the context in which it is used. For example, in some equations, G may have dimensions of length squared over mass times time squared, while in others it may have dimensions of energy times length squared divided by mass squared.

Are there any consequences if the dimensions of G are incorrect?

Yes, if the dimensions of G are incorrect, it can lead to inconsistencies and inaccuracies in the equation. This can result in incorrect predictions and conclusions, which can have significant consequences in scientific research and practical applications.

How do you ensure that the dimensions of G are correct in an equation?

To ensure that the dimensions of G are correct, it is important to carefully check the units of all the terms in the equation. Additionally, it is essential to use the correct values for G, as it may vary depending on the units used (e.g. SI units vs. Imperial units). Furthermore, peer review and validation by other scientists can help identify any potential errors in the dimensions of G.

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