Angular Momentum of the earth at the equator

[vesperaka]

Homework Statement

For a 98 kg person standing at the equator, what is the magnitude of the angular momentum about Earth's center due to Earth's rotation? Assume the Earth has a radius of 6.4 x 106 m.

Homework Equations

Angular Momentum = rmv * sin(theta)

The Attempt at a Solution

r= 6.4*10^6 m
m = 98kg
Vangular (of earth) = 7.27*10^5 rad/s

I multiplied these 3 values together and got 45597.44 kgm^2/s as my answer but it's wrong. Do I have to incorporate anything about the sin of the angle into my equation (I figured it would just be sin(90) = 1 though...)? I don't know what else to do :\

 

[alphysicist]

Hi vesperaka,

Homework Statement

For a 98 kg person standing at the equator, what is the magnitude of the angular momentum about Earth's center due to Earth's rotation? Assume the Earth has a radius of 6.4 x 106 m.

Homework Equations

Angular Momentum = rmv * sin(theta)

The Attempt at a Solution

r= 6.4*10^6 m
m = 98kg
Vangular (of earth) = 7.27*10^5 rad/s

This number does not look right to me. You have units of radians per second, and so the angular velocity of the Earth is 7.27*10^(-5) rad/s.

But this is not the v that is in your equation (it does not have the same units, for example). So you need to either find the speed v from this angular velocity, or use an equation for the angular momentum that has the angular velocity in it.

 

[baba89]

The magnitude of the angular momentum of the Earth at the equator can be calculated using the formula L = Iω, where I is the moment of inertia and ω is the angular velocity. In this case, the moment of inertia can be approximated as I = 2/5 * mr^2, where m is the mass of the Earth and r is the radius.

Plugging in the values given, we get:

I = 2/5 * (5.97 x 10^24 kg) * (6.4 x 10^6 m)^2 = 9.43 x 10^37 kgm^2

ω = 7.27 x 10^-5 rad/s (approximately)

Therefore, the magnitude of the angular momentum is:

L = (9.43 x 10^37 kgm^2) * (7.27 x 10^-5 rad/s) = 6.86 x 10^33 kgm^2/s

It is important to note that the sin(theta) term in the angular momentum formula is only necessary if the object is moving at an angle to its axis of rotation. In this case, the person standing at the equator is rotating in the same direction as the Earth's rotation, so the angle between the two is 90 degrees and the sin(theta) term is equal to 1. Therefore, it is not necessary to incorporate it into the calculation.