On the planet Newtonia, a simple pendulum having a bob with a mass of 1.26 kg and a length of 185.4 cm takes 1.59 s , when released from rest, to swing through an angle of 12.4 ∘, where it again has zero speed. The circumference of Newtonia is measured to be 5.16×104 km . What is the mass of the planet Newtonia in kg?
from given information we can see that, it is released from rest and after swinging 12.4 deg, again it's speed is zero, which means it has completed half cycle.m
Now given that time taken to complete half-cycle = 1.59 sec
So, time period of pendulum, T = 2*1.59 = 3.18 sec
Now time period of a pendulum is given by:
T = 2*pi*sqrt (L/g)
g = gravitational acceleration on planet = ?
L = length of pendulum = 185.4 cm = 1.854 m
So,
g = L*(2*pi/T)^2
g = 1.854*(2*pi/3.18)^2
g = 7.237 m/sec^2
Now for a planet 'g' is calculated by
g = G*Mp/Rp^2
Mp = mass of planet = ?
G = gravitational constant = 6.67*10^-11
Rp = Radius of planet
Given that circumference of planet = 5.16*10^4 km = 5.16*10^7 m = 2*pi*Rp
Rp = 5.16*10^7/(2*pi) = 8.212*10^6 m
So,
Mp = g*Rp^2/G
Mp = 7.237*(8.212*10^6)^2/(6.67*10^-11)
Mp = mass of planet = 7.32*10^24 kg
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