Uniform Circular Motion

Bashyboy

I am reading about uniform circular motion currently, but I am sort of confused by this paragraph:

"We next determine the magnitude of the centripetal centripetal (radial) acceleration, a_r. Because CA in fig. 5-2a is perpendicular to V₁, and CB is perpendicular to V₂, it follows that the angle Δθ, defined as the angle between CA and CB, is also the angle between V₁ and V₂. Hence the vectors V₁, V₂, and ΔV in fig. 5-2b form a triangle that is geometrically similar to the triangle CAB in fig. 5-2a"

I have attached the two figures. Why are the two triangles similar?

Doc Al

You didn't attach the figures, but I think I know what you're asking. The two triangles are isosceles triangles with the same angle between the equal sides, thus they must be similar.

This discussion may help: Centripetal Acceleration

Bashyboy

Oh, gesh, I am sorry. Thank you for the link.

Bashyboy

I have another question regarding this same topic. I can see mathematically that the radius effects centripetal acceleration, but I was wondering if there was any physical explanation to accompany this mathematical one? Thank you

Doc Al

You can think of centripetal acceleration as the rate of change of the direction of an object's velocity. And the smaller the radius, the harder it is to change direction at a given speed. Compare driving through a sharp turn (small radius) versus a gentle turn (large radius). Clearly it requires less force to navigate the gentle turn.

Does that help?

Bashyboy

Yeah, I think that does. Thank you very much.

Bashyboy

I am now reading an example problem concerning circular motion. The book says at the topmost part of the circular path, there no tension force required--that is, F_T = 0.
It says that that gravity provides the centripetal acceleration. Are the reasons why an object won't fall straight downward due to momentum and there being a horizontal component pushing it forward, back onto the circular path?

Doc Al

If you swing something--like a ball on a string--in a vertical circle with the minimum speed that still keeps the string taut, then the tension force will be zero at the top. Gravity provides all the centripetal force needed. Swing it faster and you'll begin to get string tension at the top.
At the top of the motion there's no horizontal component of force. Realize that force causes a change in motion. The change needed at the top is to keep it turning in a circle, and that requires a downward force. It doesn't fall because it is moving sideways. (Just like the moon doesn't fall into the earth, even though there is gravitational attraction pulling it.) If it moves too slow, then it will start to fall and the string will go slack.

Bashyboy

I have another question regarding circular motion, and I was not sure whether I should ask it in this thread, seeing that it is a bit older--but I'll ask here anyways. I am reading, from a physics website, about circular motion and they give proof that acceleration is directed towards the center of a circle by giving a description of a scenario with an accelerometer. This accelerometer consists of a cork tied a string, with the other end of the string tied to the lid of a capsule, which is full of water. When the accelerometer it is placed on a 2 x 4, and the 2 x 4 is spun around on a platform, the cork will lean towards the center. Now, in a car, we are similar to this cork, but we don't lean towards the center. Is this due to us having greater inertia than the cork?

Whovian

In a car, the car, not us, is experiencing an acceleration. While it pushes us along its trajectory, while this isn't very rigorous, our bodies will "try" to keep going along a straight path as predicted by Newton's First Law, and so we'll be leaning outwards. If there's confusion, it's partially because the car going around in circles isn't an inertial reference frame.

Doc Al

It has to do with the capsule being filled with water. The cork is less dense than water, so it behaves opposite to how you would behave in a car. Imagine a helium filled balloon on a string in the car--that balloon would act just like the cork in the water-filled capsule.