Calculating equatorial velocity

[awygle]

I'm confused about some results I've been getting. I tried calculating the equatorial velocity of Earth by the equation $$R\varpi=v$$, and I got ~465 $$m/s$$. According to various resources, this is correct. But then I tried to calculate it again, using $$g=a_{c}=\frac{v^{2}}{R}$$, and I got ~8000 $$m/s$$.

Obviously, the second is wrong. My question is why? My only guess is that maybe g doesn't work as $$a_{c}$$ in this case due to the normal force from the Earth or something of the sort, but that seems a weak explanation...

 

[Stonebridge]

It's wrong because the equation is not the correct one to use.
You have calculated the speed an object would have to travel in orbit around the Earth if its path was literally just above the surface. In other words, its centripetal force was provided by its weight.

 

[SAINATHAN]

earth has both rectilinear and rotatory motion.but you take a=v^2/r.so your assumption is wrong.

 

[SAINATHAN]

hi,friend in your equation how we have take a=g.that is for surface on Earth and although 'g' is not constant at all.

 

[PJC01]

First of all, it's great that you are questioning and double-checking your calculations. This is an important practice in science and can help identify any errors or misunderstandings.

In this case, the discrepancy between the two calculated values for equatorial velocity is likely due to the fact that the equations you used are based on different assumptions and principles.

The first equation, R\varpi=v, is based on the principle of conservation of angular momentum. It assumes that the Earth is rotating at a constant rate and that the distance between the Earth's axis of rotation (R) and the equator (varpi) is also constant. This equation gives us the correct value for equatorial velocity because it takes into account the Earth's rotation and the distance from the axis of rotation.

The second equation, g=a_{c}=\frac{v^{2}}{R}, is based on the principles of centripetal force and acceleration. It assumes that the Earth is a perfect sphere with a uniform mass distribution and that gravity is the only force acting on an object on the Earth's surface. However, the Earth is not a perfect sphere and has an uneven mass distribution, which means that the acceleration due to gravity (g) varies slightly at different points on the Earth's surface. Additionally, the normal force from the Earth's surface also plays a role in determining the acceleration of an object on the Earth's surface. These factors can affect the value of g and thus, the calculated equatorial velocity.

In summary, the discrepancy between the two calculated values is due to the different assumptions and principles used in the equations. It is important to consider the underlying principles and assumptions when using equations to calculate physical quantities.