Calculate the Orbital Period for the following: Earth's orbit to the Sun. I get stuck towards the end. I must use Newton's second law and have the Universal Law of Gravitation equal the Centrifugal Force.
Newton's gravitational constant: G= 6.67*10^-11 Nm^2/kg^2
Mass of Sun = 1.98*10^30 kg
Mass of the Earth 5.97*10^24 kg
Distance of the Earth from the Sun: 149.6*10^6
T = time
Centrifugal Force = (mv^2)/r
Force of Gravity = (GMm)/d^2
Mm being mass, and G/d^2 being acceleration.
The Attempt at a Solution
I multiplied Newton's gravitational constant by the mass of the Earth by the mass of the Sun, and then divided it all by the distance of the Earth from the Sun. I got 5.270280882*10^35 km. This was for the gravitational pull.
For the Centrifugal Force, I multiplied the mass of the Earth by v=((2(pi)(r))^2)/t^2. And put it over 149.6*10^6 km. I got (3.52229083*10^34 kg*km)/t^2.
I then set my two results equal to each other. Next, I multiplied by t^2 as that is the variable I am trying to find. This gave me 5.270280882*10^35 km*t^2 = 3.5229083*10^34 kg*km. Then, I divided by km on both sides to cancel it. I now have 5.270280882*10^35 t^2 = 3.5229083*10^34 kg. If I divide by the number on the left side of the equation and then square root both sides, I get a number than can't possibly be the Orbital Period of the Earth.
Can you tell where I am going wrong? Any advice is appreciated.