What is the mass of the planet Newtonia? Please help! emergency! HW DUE soon :(?

Question: since when is mass measured in kilometers?

Anyway, mass is unimportant to the swing period of a pendulum, only the pendulum's effective length and the local gravitational field.

Formula for swing period of the pendulum:

T = 2*Pi*sqrt(L/g)

The angle given has some importance, all that matters is that it is small, insignificant compared to 90 degrees. Had it been comparable to 90 degrees, our problem would be significantly more difficult to solve.

We weren't given full period, only half period. Thus:

t = Pi*sqrt(L/g)

Solve for g:

g = L*t^2/Pi^2

What causes the gravitational field? The mass of the planet Newtonia. Newton's law of gravitation will make us a relation.

g = G*M/R^2

Solve for M:

M = g*R^2/G

Substitute expression for g:

M = L*t^2*R^2/(G*Pi^2)

We weren't given R, but instead circumference C:

C = 2*Pi*R

R = C/(2*Pi)

Substitute:

M = L*t^2*C^2/(4*Pi^4*G)

Data:

L:=1.869 meters; t:=1.49 sec; C:=5.16e7 m; G:=6.673e-11 N-m^2/kg^2;

Result:

M = 4.249*10^23 kg

7.1% of Earth's mass