Why satellite that orbit earth has $F_{\rm net}$ = non-zero, if he don't accelerate?

Question

From inertial frame car in turn has centripetal force $Fs=mv^2/r$ and satellite orbit earth has $mg=mv^s/r$.

Non inertial frame

In both case gravity cancel centrifugal force,$F_{\rm net}$ =O, a=O, car accelerometer=wrong, satellite accelerometer=OK

Inertial frame

They both have Fnet=non zero, which implies that both accelerate. car accelerometer=OK, satellite accelerometer=wrong

(Accelometer show correct result in each frame only for one case)

Why satellite in inertial frame has $F_{\rm net}$ non-zero, if he don't accelerate? Why is seems that Newton 2law is not valid here?

Free body diagram for inertial frame

Answer 1

In the satellite, gravity is providing the only force on the objects. This means everything is (from an inertial view) accelerating toward the earth at a rate of whatever $g$ is at that altitude.

Since gravity is pulling on the satellite and on the internal mechanism of the accelerometer equally, the accelerometer cannot detect any displacement (just like an astronaut in the center of the satellite would not be displaced) and the instrument reads zero.

In the car it is possible to go over a hill and have the accelerometer show 0 (freefall), but you say this isn't happening.

Therefore the car has additional forces besides gravity. In this case the normal force from the road is pushing on the car (and from there, the frame of the accelerometer).

Because the normal force from the road is transmitted to the frame of the accelerometer, the frame and the test mass have different accelerations momentarily. As the test mass is displaced within the frame, the sensor within is deflected and the device shows a non-zero acceleration.

If the car were going fast enough (and the hill steep enough) that the ground were no longer pushing on the car, then the accelerometer would read zero.

Answer 2

The working principle of an accelerometer is shown here for example.

A useful property is exactly that it doesn't take our comfortable rest position in a sofa as an inertial frame. That is why a cell phone can shift from portrait to landscape as we tilt it. The internal accelerometer, based on tiny difference of capacitance of small components with cantilever assembly, are capable to detect the direction to the centre of the Earth, source of the gravitational force.

In reality it detects force and takes seriously the second law of Newton, relating it to acceleration. When it is done, there is no way but to accept that a free fall is the actual inertial frame - no force on the cantilever means zero acceleration. (the picture of a cell phone doesn't change from portrait to landscape if it thrown up rotating).

Answer 3

The two cases are different. Take a car rolling along a road that encompasses a great circle of the Earth (e.g. the equator, if such a road could be built). We will assume that the car is not accelerating $along$ the road (speedometer gives a constant reading). We will also neglect the rotation of the Earth in this example.

From the viewpoint of an outside inertial observer, the car will necessarily follow a curved path to match the curvature of the Earth. The equation of motion in the radial direction is:

$$\Sigma F=ma_r$$

$$N-W=-m\frac{v^2}{r}$$

or expressing weight using Newton's law of gravitation:

$$\frac{GM}{r^2}-\frac N m=a_r$$

Here, the normal force $N$ and the weight $W$ do not sum to zero, and there is a non-zero acceleration $a_r$ that depends on the speed $v$ of the car. Note that if the car's speed is reduced to $v=0$, the acceleration is also zero (and the accelerometer placed in the car would record 0 g).

Now consider the case of a satellite in circular orbit. In the inertial frame there is only $one$ force acting: the weight.

$$\Sigma F=ma_r$$

$$-W=-m\frac{v^2}{r}$$

The acceleration $a_r$ in this case is then, per Newton's law of gravitation:

$$m \frac{GM}{r^2}=ma_r$$

$$\frac{GM}{r^2}=a_r$$

or, for a radius negligibly different than the Earth's surface, $a_r= 1$ g (downward).

This is what an accelerometer in the satellite will read.

These are the accelerometer readings you will get if you tare the device so it reads 0 g while sitting unmoving on the ground, and reads 1 g (downward) when in free fall. This convention is in line with Newtonian mechanics. You can also tare the device to read 0 g in free fall and 1 g (upward) while stationary on the ground, which is equivalent but more in line with General Relativity. But Relativity is not necessary to understand or predict the motions outlined here.

I encourage you to draw the free body diagrams and try to derive these equations as shown.