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tahayassen
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You assume that T is constant. Rethink that.tahayassen said:v=2*pi*radius/T
As the radius increases, the speed also increases.
Doc Al said:You assume that T is constant. Rethink that.
No it doesn't.tahayassen said:It stays constant in order to maintain uniform circular motion.
Your formula was for linear velocity, not angular velocity. And no, the angular velocity is not constant as the radius changes.tahayassen said:Shouldn't angular velocity always be constant no matter what the radius is?
Doc Al said:No it doesn't.
Try getting back to basics. What force holds the satellite in orbit? Apply Newton's 2nd law.
Doc Al said:Your formula was for linear velocity, not angular velocity. And no, the angular velocity is not constant as the radius changes.
Excellent. Now realize that Gm = (v^2)r must be constant. That's what you need.tahayassen said:(mv^2)/r = (Gmm)/r^2
v^2 = Gm/r
Yes.Yes, but 2PI/T is the formula for angular velocity, correct?
For a given radius the angular velocity will be constant, but not if you change the radius.In order to maintain uniform circular motion, how can the angular velocity not be constant?
Sure it does. The formulas are derived for the case of uniform circular motion.tahayassen said:edit: From this formula, I can see that as the radius increases, the velocity goes down, but this doesn't account for uniform circular motion.
Doc Al said:For a given radius the angular velocity will be constant, but not if you change the radius.
Why do you think this? What's your reasoning?tahayassen said:Even if you change the radius, the angular velocity will stay the same.
angular velocity=2PI/T
You can change the radius as much as you'd like, but the angular velocity will stay the same.
This is a question about satellites in orbit. You can't very well ignore gravity.I can see why v^2 = Gm/r would work with large bodies that involve large gravitational forces, but what if we assumed gravity is negligible?
You can always use that for uniform circular motion. But it doesn't tell you much.Then, I guess we can use:
v=2*PI*radius/T
You merely assume that angular velocity is constant. Why?And from that we can see, since angular velcoity is constant (2PI/T), as you increase the radius, the velocity also increases.
For a given orbital radius, the angular velocity is constant. That's what uniform circular motion means. But when you change the radius, the angular velocity changes to a new value. So you cannot assume that angular velocity remains constant for different radii.tahayassen said:angular velocity = 2PI/T
Because angular velocity comes from multiplying 2 times PI divided by T, the radius has no effect on the angular velocity.
Assume the radius is 5 m and the period is 10 seconds.
angular velocity 1 = 2PI/10
Assume the radius is 10 m and the period is 10 seconds.
angular velocity 2 = 2PI/10
angular velocity 1 = angular velocity 2
Doc Al said:For a given orbital radius, the angular velocity is constant. That's what uniform circular motion means. But when you change the radius, the angular velocity changes to a new value. So you cannot assume that angular velocity remains constant for different radii.
Sure, if angular velocity remains the same. But it doesn't!tahayassen said:You can also think of it logically.
To maintain the same angular velocity, if you increase the radius of anything, it will need to cover the larger amount of arc length over the same amount of time. Therefore, the velocity must increase, to maintain the same amount of time taken.
That's the definition of uniform circular motion. But all that means is that for a given radius, the velocity will be constant. (As opposed to having a changing speed.) Of course, for a satellite that constant velocity depends on the radius of the orbit!tahayassen said:Alright, so you agree that when there is uniform circular motion, the angular velocity is constant?
Doc Al said:For a given orbital radius, the angular velocity is constant. That's what uniform circular motion means. But when you change the radius, the angular velocity changes to a new value. So you cannot assume that angular velocity remains constant for different radii.
For a given orbit (radius), both linear speed and angular speed are constant. (At least in uniform circular motion.)tahayassen said:So in other words, you are saying that linear speed is constant? But angular speed is not constant?
v = 2*PI*r/T
If linear speed is the same, then when you increase the radius, the period also increases.
No, that method is wrong. You cannot assume that centripetal acceleration stays the same!tahayassen said:Alright, I think I understand. This is how my teacher told me the answer was:
She assumed that centripetal acceleration stayed the same.
a1 = a2
(v1)^2/r1 = (v2)^2/r2
She said that as r2 increases, v2 must decrease. Does that method work too?
If that equation were true it would imply that as r increased so must v. Just the opposite of what she said.tahayassen said:a1 = a2
(v1)^2/r1 = (v2)^2/r2
She said that as r2 increases, v2 must decrease. Does that method work too?
Uniform circular motion is a type of motion in which an object moves along a circular path at a constant speed. This means that the object covers the same distance in the same amount of time, and its velocity remains constant throughout the motion.
The main difference between uniform circular motion and simple harmonic motion is the type of path the object follows. In uniform circular motion, the object moves along a circular path, while in simple harmonic motion, the object moves back and forth along a straight line.
In uniform circular motion, the velocity and acceleration of the object are always perpendicular to each other. This means that the magnitude of the acceleration remains constant, but its direction changes as the object moves along the circular path. The velocity, on the other hand, remains constant in magnitude and direction.
Centripetal force is the force required to keep an object moving in a circular path. In uniform circular motion, the centripetal force is directed towards the center of the circle and is equal to the mass of the object multiplied by its centripetal acceleration.
Some common examples of uniform circular motion include a satellite orbiting the Earth, a car moving around a roundabout, and a spinning top. Any object that moves along a circular path at a constant speed can be considered an example of uniform circular motion.