Centrifugal force/ pseudo force in circular motion [closed]

Question

As an example, I have considered earth's circular motion around the sun and an observer revolving around the sun too, both at different distances $r_2$ and $r_1$ respectively. The angular velocity for both is the same, say $\omega$.

So with respect to the observer, there are two forces on the earth, first the centripetal force $Mr_2\omega^2$ and the other is the pseudo force due to the acceleration of observer which is $Mr_1\omega^2$. ($M$ is mass of earth)

Now, the observer would see the earth at rest since their angular velocities are same. This implies that magnitude of centripetal force is equal to the magnitude of pseudo force or $Mr_1\omega^2=Mr_2\omega^2$. But this can't be true as then $r_1$ will be equal to $r_2$ which is absurd.

Answer 1

the other is the pseudo force due to the acceleration of observer which is M(r1)w². (M is mass of earth)

Here is the mistake. The pseudo force is $M r_2 \omega^2$. The magnitude of the pseudo force on an object depends on that object’s position in the non-inertial frame. Not on the position of the observer.

Answer 2

... the observer would see the earth at rest since their angular velocities are same ...

Their angular velocities about the Sun are the same. However, if the observer is revolving around the Sun but not rotating, then the observer sees both the Earth and the Sun revolve around them with angular velocity $\omega$. Since the earth is a distance $r_2-r_1$ from them they infer that there must be a net centripetal force $M(r_2-r_1)\omega^2$ acting on the Earth. They know that the gravitational force acting on the Earth is $Mr_2\omega^2$ towards the Sun. So they infer the existence of a pseudo force $Mr_1\omega^2$ acting away from themselves.

If, on the other hand, the observer is revolving around the Sun and rotating so that the Earth and the Sun appear to stand still, then the pseudo force depends on distance from the observer (as another answer points out). If an object of mass $M$ is at a distance $r$ from the observer then the pseudo force is $Mr_1\omega^2$ away from the Sun due to revolving around the Sun plus $Mr\omega^2$ away from the observer due to the observer's rotation. For the Earth at distance of $r=r_2-r_1$ from the observer we have

$Mr_1\omega^2 + M(r_2-r_1)\omega^2 = Mr_2\omega^2$

as expected.

Answer 3

So with respect to the observer, there are two forces on the earth, first the centripetal force $Mr_2\omega^2$ and the other is the pseudo force due to the acceleration of observer which is $Mr_1\omega^2$

In cases like this, where and object appears to be stationary in an accelerating reference frame, the pseudo centrifugal force is equal in magnitude and opposite in direction to the centripetal force and we use the radial distance of the object ($r_2$) from the centre of rotation to determine the centrifugal force and not the radial location of the observer.

As, Biophysicist points out, the centrifugal force observed in a rotating reference frame is not always a reaction to an opposite centripetal force. For example, if an observer is standing on the perimeter of a rotating turntable, and releases a ball, the ball will appear to accelerate outward in a curved path and the observer will attribute this to the pseudo centrifugal force and in this case the centrifugal force appears to be present even without the corresponding centripetal force. In this latter case the a n accelerometer mounted on the ball would measure no proper acceleration and the ball is actually moving inertially.

This pseudo force is usually inferred and is not directly measurable. When cornering hard in a car we can feel radial forces acting on us. The force we feel is the inward acting centripetal force and an accelerometer measures only this inward force. For a long time it was assumed that centrifugal forces provided the force required to counter the centripetal force of gravity acting on an orbiting body. When Einstein formulated General Relativity, gravity was no longer considered a force and orbiting bodies just follow geodesics. Since there is no centripetal force provided by gravity, there is no centrifugal force required to balance it in the case of orbiting bodies. An accelerometer attached to a small satellite would not measure any proper radial forces. (This is why astronauts appear to be weightless in International Space Station.) So for your case of the Earth orbiting the Sun, there is actually no centrifugal force acting on the Earth as a result of it orbiting the Sun. (but there is centrifugal force experienced by a non inertial observer standing on the the surface of the Earth as a result of Earth's own spin and this slightly reduces the apparent force of gravity and gives the Earth its oblate shape).