Circular motion and acceleration

[Amr Elsayed]

Hello,
Here is a thing I'm confused about.
when I first knew about centripetal force and satellites. I thought of it the exact way like projectiles. I thought like :satellites do fall. If we specify a single direction, the magnitude of the velocity is really changing. I thought of its path that it's a combination between inertia outwards and gravity to the center, and that inertia overcomes gravity, so the satellite has this kind of a circular path, and the direction of motion by the 2 forces is closer to the direction of inertia pushing outwards.
I have read on the physics classroom that magnitude of velocity is not changing, but since direction is changing, a change in velocity leads to acceleration. But acceleration is also a vector quantity and it should have a direction !
It really makes no sense for me to say that direction of this acceleration is inwards
Please help me find out what's wrong right here
Regards

 

[Orodruin]

Since the satellite is moving around, the "inwards" direction is changing and therefore also the direction of the acceleration.

 

[Amr Elsayed]

Since the satellite is moving around, the "inwards" direction is changing and therefore also the direction of the acceleration.

But as far as I know, acceleration is a changing in velocity in a specified direction, right ? But what does it mean that there is an acceleration inwards in that case ?
and what about

satellites do fall. If we specify a single direction, the magnitude of the velocity is really changing. I thought of its path that it's a combination between inertia outwards and gravity to the center, and that inertia overcomes gravity, so the satellite has this kind of a circular path, and the direction of motion by the 2 forces is closer to the direction of inertia pushing outwards.

Regards

 

[Orodruin]

Consider a vector of fixed magnitude but which changes direction in time. The tip of the vector is then always on a circle and so any small change in the vector must be tangential to the circle. Since the vector points radially, the direction tangential to the circle must be orthogonal to the vector. This is why the acceleration is orthogonal to the velocity if the speed is constant.

Mathematically, fixed speed means that $d\vec v^2/dt= 2\vec v \cdot d\vec v/dt = 2 \vec v \cdot \vec a = 0$, ie, the velocity and acceleration are orthogonal.

 

[lavinia]

It really makes no sense for me to say that direction of this acceleration is inwards

If the acceleration had a component tangent to the circle then the speed of the satellite would accelerate by the magnitude of that component. But the speed is constant so any acceleration must be orthogonal to the velocity vector. This is true of any constant speed curve.

 

[SUMIT16]

there are two accelerations here one inwards called 'centripetal' and other 'tangential'. here tangential accln is 0. so actually the centripetal acceleration ie inwards is actually responsible for the change in direction, hence we say velocity is not constant as the direction is changing.
so for any curved motion the centripetal acceleration is the reason behind the change in the direction of motion.

 

[A.T.]

But as far as I know, acceleration is a changing in velocity in a specified direction, right ? But what does it mean that there is an acceleration inwards in that case ?

Exactly what you said, the change of the velocity vector. See bottom of this page:
http://hyperphysics.phy-astr.gsu.edu/hbase/cf.html

 

[Amr Elsayed]

But the speed is constant so any acceleration must be orthogonal to the velocity vector

o for any curved motion the centripetal acceleration is the reason behind the change in the direction of motion

Exactly what you said, the change of the velocity vector

Okay, so it's correct to think of satellites like projectiles. the only difference is that projectiles doesn't have enough speed for their curvature to match the one of Earth ? since projectiles are accelerating downward because of gravity, it's the same with satellites that are falling on the side of earth, so acceleration is inwards the same way ? but in projectiles acceleration is in one direction no matter how much the path is curved, this seems a bit different from satellite's case for me

 

[Orodruin]

Okay, so it's correct to think of satellites like projectiles. the only difference is that projectiles doesn't have enough speed for their curvature to match the one of Earth ? since projectiles are accelerating downward because of gravity, it's the same with satellites that are falling on the side of earth, so acceleration is inwards the same way ? but in projectiles acceleration is in one direction no matter how much the path is curved, this seems a bit different from satellite's case for me

In the projectile case, you are typically considering distances which are so small that the gravitational field may be considered homogeneous. This is not the case for satellites. The change in the gravitational field is not large enough to be noticeable. In general, a satellite would not have a circular orbit either. Newton's laws of motion result in orbits which are elliptical. For these orbits, the acceleration is not perpendicular to the direction of motion.

 

[Amr Elsayed]

In the projectile case, you are typically considering distances which are so small that the gravitational field may be considered homogeneous. This is not the case for satellites. The change in the gravitational field is not large enough to be noticeable. In general, a satellite would not have a circular orbit either. Newton's laws of motion result in orbits which are elliptical. For these orbits, the acceleration is not perpendicular to the direction of motion.

That means the velocity of the satellite changes due to attitude, it changes every time there is a difference between its curvature and curvature of earth" when centripetal force and path of the satellite are not orthogonal " but this has no contradiction with what I said, right ?
You explained here before why acceleration needs to be orthogonal not to affect speed. but I'm not sure that I get what orthogonal acceleration means. it is always changing in direction, that's why the satellite just moves without getting pulled to that location on earth, right ?

 

[A.T.]

Okay, so it's correct to think of satellites like projectiles. the only difference is that projectiles doesn't have enough speed for their curvature to match the one of Earth ?

Yes:
https://en.wikipedia.org/wiki/Newton's_cannonball

but in projectiles acceleration is in one direction no matter how much the path is curved, this seems a bit different from satellite's case for me

There is no difference, Both are accelerated towards the center of the Earth.

I'm not sure that I get what orthogonal acceleration means. it is always changing in direction

Yes, it is.