Respuesta :
Answer:
The mass of the planet Newtonia is [tex]9.07\times 10^{26}\ kg[/tex].
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:
[tex]T=2\pi \sqrt{\dfrac{l}{g}}[/tex]
g is the acceleration due to gravity on planet Newtonia
[tex]g=\dfrac{4\pi^2l}{T^2}[/tex]
For one complete cycle, T = 2.84 s
[tex]g=\dfrac{4\pi^2\times 1.85}{(2.84)^2}[/tex]
[tex]g=9.05\ m/s^2[/tex]
The formula of acceleration due to gravity on any surface is given by :
[tex]g=\dfrac{GM}{r^2}[/tex]
Since, [tex]2\pi r=514000\times 10^3[/tex]
[tex]r=8.18\times 10^7\ m[/tex]
[tex]M=\dfrac{gr^2}{G}[/tex]
[tex]M=\dfrac{9.05\times (8.18\times 10^7)^2}{6.67\times 10^{-11}}[/tex]
[tex]M=9.07\times 10^{26}\ kg[/tex]
So, the mass of the planet Newtonia is [tex]9.07\times 10^{26}\ kg[/tex]. Hence, this is the required solution.
The mass of the planet Newtonia that has a circumference of 51,4000 km is; M = 9.085 × 10²⁶ kg
We are given;
Mass of the bob; m = 1.25 kg
Length of the simple pendulum; l = 185 cm = 1.85 m
Half Period; T_half = 1.42 s
One full period; T = 1.42 × 2 = 2.84 s
Circumference of Newtonia; C = 51,4000 km = 514 × 10⁶ m
Thus;
2πr = 514 × 10⁶
r = (514 × 10⁶)/2π
r = 81.806 × 10⁶ m
Formula for period of the simple pendulum is given by;
T = 2π√(l/g)
where g is acceleration due to gravity on newtonia
Making g the formula gives;
g = 4π²l/T²
Plugging in the relevant values gives;
g = 4π² × 1.85/2.84²
g = 9.055 m/s²
Now, formula for acceleration due to gravity on the planet surface will be;
g = GM/r²
where;
g is gravitational constant = 6.67 × 10⁻¹¹ N.m²/kg²
m is mass of planet newtonia
9.055 = 6.67 × 10⁻¹¹ × M/(81.806 × 10⁶)²
M = 9.055 × (81.806 × 10⁶)²/(6.67 × 10⁻¹¹)
M = 9.085 × 10²⁶ kg
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