Movement caused by Centripetal Force

[Robzoid]

I've been thinking about centripetal force and circular motion. I know that an object experiencing circular motion has a velocity vector tangential to the circular path and an acceleration vector perpendicular to its motion, pointing towards the center of the circle.

What I don't get is why doesn't the inward acceleration cause the object to move radially inward towards the center of the circle, like a spiral. How is it that radially inward acceleration doesn't always result in radially inward motion? Any explanation will be greatly appreciated.

 

[Dale]

How is it that radially inward acceleration doesn't always result in radially inward motion?

Do you drive? If the road turns to the left (no banking) then you need to turn to the left just to stay in your lane. If you want to change lanes to the right then you don't actually turn right, you just turn left a little less. So there is some force required just to stay in your lane on a curve. You can actually move outward even with radially inward force

 

[PeroK]

I've been thinking about centripetal force and circular motion. I know that an object experiencing circular motion has a velocity vector tangential to the circular path and an acceleration vector perpendicular to its motion, pointing towards the center of the circle.

What I don't get is why doesn't the inward acceleration cause the object to move radially inward towards the center of the circle, like a spiral. How is it that radially inward acceleration doesn't always result in radially inward motion? Any explanation will be greatly appreciated.

Here are three possible answers:

1) Do the maths!

2) It's a balance. If you have a particle moving at a certain speed and you want it to move in a circle about a given central point, then you must apply a force of precisely the right magnitude. And you must ensure the force changes direction so that it always points at the centre.

If you apply too great a force, the particle will spiral inward; too little force and it will spiral outward. The force must be just right to get circular motion about your given centre.

Note that the particle will spiral outward or inward until the distance from your central point is just right for the applied force and then it will stay in a circle with the increased or decreased radius.

3) Think of a ball on a string, fixed to a central point and give the ball a push. The tension in the string will automatically adjust to be just right to keep the ball in a circle. The ball is continuously trying to move in a straight line, but the constant tension is just enough to keep it moving in a circle.

 

[A.T.]

How is it that radially inward acceleration doesn't always result in radially inward motion?

https://en.wikipedia.org/wiki/Centripetal_force#Calculus_derivation

 

[rcgldr]

Assuming constant speed, as centripetal force increases, the radius of curvature decreases, but the direction of the centripetal force changes as the car turns, and the lines of centripetal forces sampled over time do not cross at a common "center" of a circle if the path is not circular.

In the attached images, hole.jpg shows an example of forces directed to a fixed point. In this case, an object sliding on a frictionless surface attached to a string that is pulled into or allowed to extend from a hole at a fixed point in the surface. When the path is not circular, part of the force from the string is in the direction of motion, causing a change in speed. If the string is pulled in, the object speeds up, and if the string is let out, the object slows down. The short lines in the image are lines perpendicular to the objects path, different than the direction of the string.

Pole.jpg shows an example of an object sliding on a frictionless surface attached to a string that wraps or unwraps around a pole. In this case, the string is always perpendicular to the objects path, so the string only exerts a centripetal force, and there is no change in speed (until the object hits the pole).

 

[Joe Dawson]

Your question should be reworded as "SPEED tangential to the circular path and a FORCE perpendicular to its motion, pointing towards the center of the circle."

Speed is a scalar. Velocity is a vector consisting of two components: speed and direction. A change in either the speed OR the direction requires force, and is called acceleration. Depending on the direction of the force, one of three things will happen:

1. If the force is applied in the same direction as the existing motion, the speed changes but the direction does not (acceleration in a straight line).
2. If the force is applied at a right angle to the existing motion, the direction changes but the speed does not (circular motion: acceleration toward the center).
3. If the force is applied in any other direction, both the speed AND the direction change (complex motion: acceleration in a curved path).

 

[PeroK]

2. If the force is applied at a right angle to the existing motion, the direction changes but the speed does not (circular motion: acceleration toward the center).

It's perhaps worth adding that in this case the centre of the circular motion is determined by the magnitude of the force. And that the force must continuously change direction to remain perpendicular to the velocity in order to achieve circular motion.