Dimensional analysis and exponents

[addedline8]

I decided to take physics this year, my senior year in high school, and my teacher has decided he isn't going to explain things before he assigns us homework. Any and all help is appreciated.

1. The period T of a simple pendulum is the amount of time required for it to undergo one complete oscillation. If the length of the pendulum is L and the acceleration of gravity is g, then T is given by:
2. T = 2 (pi) L^p g^q
3. T = 2 (pi) L^p (L/T²)^q

Find the powers p and q required for dimensional consistency.

 

[bp_psy]

Hint. L is in meters, g is in meter/second^2 and period T is in seconds. Now just do the algebra.

 

[PeterO]

I decided to take physics this year, my senior year in high school, and my teacher has decided he isn't going to explain things before he assigns us homework. Any and all help is appreciated.

1. The period T of a simple pendulum is the amount of time required for it to undergo one complete oscillation. If the length of the pendulum is L and the acceleration of gravity is g, then T is given by:



2. T = 2 (pi) L^p g^q



3. T = 2 (pi) L^p (L/T²)^q

Find the powers p and q required for dimensional consistency.

keep going the way you have started..

T = 2 (pi) L^p (L/T²)^q

T = 2 (pi) L^p L^q/T^2q

T = 2 (pi) L^p+q.T ^-2q

Now compare dimensions left and right.

What must -2q equal?

What must p+q equal?

 

[addedline8]

Thanks a lot guys. I appreciate the help.

 

[rwisz]

I understand that dimensional analysis is a crucial tool in physics to ensure that equations and calculations are mathematically consistent and physically meaningful. It involves breaking down physical quantities into their fundamental units and using them to determine the powers of each variable in an equation.

In the given equation for the period of a simple pendulum, we can see that the length L is measured in meters (m) and the acceleration of gravity g is measured in meters per second squared (m/s^2). Therefore, the powers p and q must be chosen in a way that the units on both sides of the equation are consistent.

For the first equation, the powers p and q should be 0 and 1, respectively, to ensure that the units on both sides are meters (m).

For the second equation, the powers p and q should be 1 and -1, respectively, to ensure that the units on both sides are seconds (s).

For the third equation, the powers p and q should be 1 and -2, respectively, to ensure that the units on both sides are seconds squared (s^2).

It is important to note that these powers are not arbitrary and are determined based on the units of the variables in the equation. I would suggest discussing this topic with your teacher to gain a better understanding of dimensional analysis and how it applies to physics equations. Additionally, there are many online resources and tutorials available that can help you practice and improve your skills in this area. Good luck with your physics studies!