# Relationship of centripetal force and period

### Centripetal force - Wikipedia

If r is the radius of the path, and we define the period, T, as the time it In circular motion many people use the term centripetal force, and say. For example, in your second equation, the centripetal force is directly Both relationships are valid, depending on which variables are being. In such cases the acceleration is sideways, towards the center, or centripetal. Centripetal force is parallel to centripetal acceleration. frequency and period.

But first, what is period and what is frequency? Well, period is how long does it take to complete a cycle?

## Centripetal Force

And if we're talking about uniform circular motion, a cycle is how long does it take, if this is, say, some type of a tennis ball that's tethered to a nail right over here and it's moving with some uniform speed, a period is, well, how long does it take to go all the way around once? So, for example, if you have a period of one second, this ball would move like this, one second, two seconds, three seconds, four seconds. That would be a period of one second.

If you had a period of two seconds, well, it would go half the speed. You would have one second, two seconds, three seconds, four seconds, five seconds, six seconds. And if you went the other way, if you had a period of half a second, well, then it would be one second, two seconds, and so your period would be half a second.

It would take you half a second to complete a cycle. The unit of period is going to be the second, the unit of time and it's typically given in seconds.

Now, what about frequency? Well, frequency literally is the reciprocal of the period. So frequency is equal to one over, let me write that one a little bit neater, one over the period.

## Connecting period and frequency to angular velocity

And one way to think about it is well, how many cycles can you complete in a second? Period is how many seconds does it take to complete a cycle, while frequency is how many cycles can you do in a second? So, for example, if I can do two cycles in a second, one second, two seconds, three seconds, then my frequency is two cycles per second. And the unit for frequency is, sometimes you'll hear people say just per second, so the unit, sometimes you'll see people just say an inverse second like that, or sometimes they'll use the shorthand Hz, which stands for Hertz.

And Hertz is sometimes substituted with cycles per second. So this you could view as seconds or even seconds per cycle. And this is cycles per second. Now with that out of the way, let's see if we can connect these ideas to the magnitude of angular velocity.

### Uniform circular motion

So let's just think about a couple of scenarios. Let's say that the magnitude of our angular velocity, let's say it is pi radians, pi radians per second.

So if we knew that, what is the period going to be? Pause this video and see if you can figure that out. So let's work through it together. So, this ball is going to move through pi radians every second. So how long is it going to take for it to complete two pi radians? Well, if it's going pi radians per second, it's gonna take it two seconds to go two pi radians.

We don't know the angle, but that's OK because we can solve for it. Adding forces in the y direction gives: This can be solved to get the angle: In the x direction there's just the one force, the horizontal component of the tension, which we'll set equal to the mass times the centripetal acceleration: We know mass and tension and the angle, but we have to be careful with r, because it is not simply the length of the rope.

It is the horizontal component of the 1.

**Centripetal Acceleration & Force - Circular Motion, Banked Curves, Static Friction, Physics Problems**

Rearranging this to solve for the speed gives: Example 2 - Identical objects on a turntable, different distances from the center. Let's not worry about doing a full analysis with numbers; instead, let's draw the free-body diagram, and then see if we can understand why the outer objects get thrown off the turntable at a lower rotational speed than objects closer to the center.

In this case, the free-body diagram has three forces, the force of gravity, the normal force, and a frictional force. The friction here is static friction, because even though the objects are moving, they are not moving relative to the turntable.

If there is no relative motion, you have static friction. The frictional force also points towards the center; the frictional force acts to oppose any relative motion, and the object has a tendency to go in a straight line which, relative to the turntable, would carry it away from the center.

So, a static frictional force points in towards the center. Summing forces in the y-direction tells us that the normal force is equal in magnitude to the weight.

### Connecting period and frequency to angular velocity (video) | Khan Academy

In the x-direction, the only force there is is the frictional force. The maximum possible value of the static force of friction is As the velocity increases, the frictional force has to increase to provide the necessary force required to keep the object spinning in a circle. If we continue to increase the rotation rate of the turntable, thereby increasing the speed of an object sitting on it, at some point the frictional force won't be large enough to keep the object traveling in a circle, and the object will move towards the outside of the turntable and fall off.