Vertical Circles

Vertical circles are circles where an object is moving such that the net vertical force on it equals the centripetal force required to keep it in a circular motion. This is usually seen when you spin a yo-yo using string tension in a vertical circle such that gravity acts on it and tension, too. Vertical circles are very important for loop-the-loops in amusement park rides as both the normal force exerted by the track on you and gravity are at play.

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In order to illustrate how vertical circles work, let's use something that you've most likely seen if not experienced: a roller coaster loop-the-loop.

As you can see in this loop-the-loop in the image, the ride will go in a vertical loop. There are 2 main forces at play(ignore the friction on the ride from the rails)on the roller coaster cart going through the loop: the normal force of the rails on the cart and the cart's weight.

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The absolute minimum speed required to at the top is given by:

Interestingly, if we take the centripetal force on the sides, since no component of the cart's weight acts centripetally, the centripetal force required to keep it in circular motion is solely supplied by the normal force on the sides.

*Please note that vertical circles can be have any 2 forces at play keeping the object in circular motion but usually gravity is one of them. However, the other one can be any force like tension(which is why you can twirl yo-yos in vertically circular motion)*

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If the cart isn't at a given side, the necessary centripetal force is given by the centripetal component of the normal force. If you draw up a triangle with one side as the radius of the circle from the center of the circle to where the cart is and you connect the cart to the center with a horizontal and vertical line, you get a right triangle. You can then use a trigonometric ratio or the Pythagorean Theorem to solve for the centripetal component of force at that point.

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The relationship between the normal force at the bottom and the normal force at the top is given by:

*This also works for forces like tension*