Question

On the planet Newtonia, a simple pendulum having a bob with a mass of 1.35kg and a length of 187.0cm takes 1.42s , when released from rest, to swing through an angle of 12.0?, where it again has zero speed. The circumference of Newtonia is measured to be 5.15

Answer 1

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:=1.87 meters; t:=1.42 sec; C:=5.15e7 m; G:=6.673e-11 N-m^2/kg^2;

Result:
M = 4.249*10^23 kg

7.1% of Earth's mass