Question

On the planet Newtonia, a simple pendulum having a bob with mass 1.25 kg and a length of 185.0 cm takes 1.42 seconds, when released from rest, to swing through an angle of 12.5 degrees, where it again has zero speed. The circumference of Newtonia is measured to be 51,4000 km. What is the mass of the planet Newtonia?

Answer 1

Answer:

The mass of the planet Newtonia is $9.07\times 10^{26}\ kg$.

Explanation:

It is given that,

Mass of the bob, m = 1.25 kg

Length of the simple pendulum, l = 185 cm = 1.85 m

Time taken by the simple pendulum, T = 1.42 s

The circumference of Newtonia is measured to be 51,4000 km. The time period of the simple pendulum is given by the formula as:

$T=2\pi \sqrt{\dfrac{l}{g}}$

g is the acceleration due to gravity on planet Newtonia

$g=\dfrac{4\pi^2l}{T^2}$

For one complete cycle, T = 2.84 s

$g=\dfrac{4\pi^2\times 1.85}{(2.84)^2}$

$g=9.05\ m/s^2$

The formula of acceleration due to gravity on any surface is given by :

$g=\dfrac{GM}{r^2}$

Since, $2\pi r=514000\times 10^3$

$r=8.18\times 10^7\ m$

$M=\dfrac{gr^2}{G}$

$M=\dfrac{9.05\times (8.18\times 10^7)^2}{6.67\times 10^{-11}}$

$M=9.07\times 10^{26}\ kg$

So, the mass of the planet Newtonia is $9.07\times 10^{26}\ kg$. Hence, this is the required solution.