Centripetal motion, find radius - which solution is correct?

[Tyrannosaurus_]

Homework Statement

The planet Saturn will orbit the Sun every 19 Earth years (= 599 184 000 seconds). Assume Saturn travels on a circular orbit with a speed of 9 690m/s. What is the radius of the orbit of Saturn?

I can solve this in two different ways, and get two different values. Which is the correct approach, & why?

2. Homework Equations & attempt at solution.
T = (2πr)/V - that is, period is 2π*radius over speed.,
so, r = (T*V)/2π
so, radius = 9.24 x 10¹¹m

However, if, the centripetal forces equals the universal law of gravitation, that is,
F_c = F_g
(m*v²)/r = (Gm₁m₂)/r²
r = (G*m_sun)/v²
r = [(6.67x10^-11)(1.99x10³⁰kg)]/(9690m/s²)
r = 1.41 x 10¹²m

So, why the difference? What applies, and what doesn't apply and why?

Thanks!

 

[ehild]

Homework Statement

The planet Saturn will orbit the Sun every 19 Earth years (= 599 184 000 seconds). Assume Saturn travels on a circular orbit with a speed of 9 690m/s. What is the radius of the orbit of Saturn?

I can solve this in two different ways, and get two different values. Which is the correct approach, & why?

2. Homework Equations & attempt at solution.
T = (2πr)/V - that is, period is 2π*radius over speed.,
so, r = (T*V)/2π
so, radius = 9.24 x 10¹¹m

However, if, the centripetal forces equals the universal law of gravitation, that is,
F_c = F_g
(m*v²)/r = (Gm₁m₂)/r²
r = (G*m_sun)/v²
r = [(6.67x10^-11)(1.99x10³⁰kg)]/(9690m/s²)
r = 1.41 x 10¹²m

So, why the difference? What applies, and what doesn't apply and why?

Thanks!

Both of your methods are correct, but the problem maker gave wrong orbital period. It is 29 years, instead of 19.

 

[Tyrannosaurus_]

Both of your methods are correct, but the problem maker gave wrong orbital period. It is 29 years, instead of 19.

Thanks so much! I was going crazy!