What is Uniform Circular Motion?

Uniform circular motion is just what it sounds. When an object moves around in a circular path at a uniform speed, it is in uniform circular motion. For example, if you held one end of a yoyo string and spun around in place while the yoyo traveled around in a circle, that yoyo is experiencing uniform circular motion. 🪀

There are many different examples of objects moving in a uniform circular motion, but they are all kept moving in a uniform circular motion by a centripetal force. A centripetal force is the sum of all the forces acting on an object that point towards the middle of that object’s circular path. ➕

Let’s identify what force(s) act as the centripetal force in our yoyo example. We draw a free body diagram with the forces acting on the yoyo and look at which force(s) or force components are pointing towards the center of the yoyo’s circular path. 🎨

The Centripetal Force and Centripetal Acceleration

How does a centripetal force make an object travel around in a circular path? Think back to Newton’s second law of motion, which tells us net force = mass times acceleration, which means that a net force causes an acceleration. In this case, our net force (pointing towards the center of the object’s circular path) is the centripetal force. 🙌

This centripetal force causes a centripetal acceleration that also points towards the center of the object's circular path! Recall that acceleration is a change in velocity in which velocity is the speed and direction an object is moving in. In other words, acceleration is a change in an object's speed, direction that it's moving in, or both. ⛽

For uniform circular motion, the centripetal acceleration just causes a shift in the object's direction (since the object is moving with a constant speed. There are situations where an object can have a centripetal acceleration, be moving in a circular path, and have different speeds, which we'll cover later). 🏹

At any given time, the object's velocity points tangentially to the circular path it's moving in. The acceleration points towards the center of the circular path. This makes the object follow the purple arrow and ultimately travel along the circular path.

Let’s think back to our yoyo example! Before the yoyo starts moving in a uniform circular motion, you must give it a horizontal force by tugging on the string. This horizontal force moves the yoyo by giving it a horizontal velocity. 🔑

Then, as you hold on to the string and provide an upward force on the yoyo, the yoyo goes up into its circular path. The velocity of the yoyo stays pointing out in the horizontal direction. ⭕

Still, now as you spin around and hold the string, the yoyo enters uniform circular motion, and the centripetal acceleration continuously changes which way the yoyo’s velocity is pointing. Despite changing directions, the velocity will always point tangentially to the circle! 🎯

Here’s the direction of the yoyo’s velocity and acceleration at different points on the yoyo’s path:

It also makes sense that the velocity is tangential to the yoyo’s circular path if we think about what happens if we let go of the string. If you let go of the string while spinning, the yoyo now has no centripetal acceleration, but it still has it’s tangential velocity. That means that the yoyo will travel in whatever direction its tangential velocity pointed at the moment you let go of the string. It will not keep traveling in a circle; it will fly forwards, instead. 🪁

Newton’s Second Law and Calculating Centripetal Acceleration

As we saw earlier, Newton’s Second Law tells us that net force = mass * acceleration. This can also be applied to centripetal force and centripetal acceleration by saying:

(or centripetal force equals mass times centripetal acceleration.) 

Another formula we use to find centripetal acceleration is:

(or centripetal acceleration equals the velocity of the object squared divided by the radius of the object’s circular path.)

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