Question

The Earth's radius is 6378.1 kilometers. A mad scientist has come up with the simultaneously awesome and terrifying plan to increase the speed of the Earth's rotation until people at the Earth's equator experience a centripetal (radial) acceleration with a magnitude equal to g (9.81 m/s2 ), eectively making them experience weightlessness. If the mad scientist succeeds in their dastardly plan, what would be the new period of the Earth's rotation?

Answer 1

Answer: T = 5068 s

Explanation:

Given

Radius of the earth, r = 6378.1 km

Centripetal acceleration, g = 9.81 m/s²

Period of rotation, T = ?

a = w²r, where

a is the centripetal acceleration

w is the angular velocity

r is the radius

9.81 = w² * 6378.1*10^3

w² = 9.81 / 6378.1*10^3

w² = 1.538*10^-6

w = 1.24*10^-3 rad/s

To get the period of the earth, we use the formula

w = 2π / T, so that

T = 2π / w

T = (2 * 3.142) / 1.24*10^-3

T = 6.284 / 1.24*10^-3

T = 5067.74 s

Therefore, the earth would rotate at a period of 5068 s or 1 hour and 24 minutes

Answer 2

Answer:

The time period of Earth’s rotation would be 84.4 minutes

Explanation:

Given that,

Centripetal acceleration = 9.81 m/s²

Radius ,r = 6378.1 km

Velocity, v

Centripetal acceleration is

$a_c=\dfrac{v^2}{r}$

$a_c=\dfrac{v^2}{r}\\\\\Rightarrow v=\sqrt{a_cr}\\\\\Rightarrow v=\sqrt{9.81\times 6378100}\\\\\Rightarrow v=7910.067\ m/s$

Time period is given by

$T=\dfrac{2\times \pi \times r}{v\times 60}\\\\ T=\dfrac{2\times \pi \times 6378.1\times 10^3}{7910.06706\times 60}\\\\ T=84.4\ minutes$

Hence, the time period of Earth’s rotation would be 84.4 minutes