David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.
After watching this video, you will be able to describe motion of an object moving in a vertical circle and derive and use equations for this motion in problem solving. A short quiz will follow.
Motion in a Vertical Circle
In another lesson, we talked about uniform circular motion, which is motion in a circle at a constant speed. In this motion, the centripetal force, the force that points towards the center of a circle, is always constant. But that isn't always the case.
If you've ever swung a shopping bag in a vertical circle, you know that it definitely doesn't feel the same at the top and the bottom of that circle. The shopping bag slows down at the top of the circle and speeds up at the bottom. The amount of force you have to apply is constantly changing, too.
This happens because gravity is pointing downwards. At the top of the circle, gravity is pointing in the same direction as the tension in the bag. At the bottom of the circle, gravity is pointing in the opposite direction to the tension. And in between it's... in between. This means that the tension force has to vary to compensate.
So how do we analyze motion in a vertical circle?
Energy equations for motion
To analyze motion in a vertical circle and produce some equations, we can use a combination of energy and forces. The energy equation for the motion is probably the easiest. At the top, we have gravitational potential energy and kinetic energy, and at the bottom, we have just kinetic energy. So one half mv-squared at the top plus mgh at the top is equal to one half mv-squared at the bottom. Where v-top is the velocity at the top in meters per second, v-bottom is the velocity at the bottom, also in meters per second, m is the mass of the object moving in a circle in kilograms, g is the acceleration due to gravity, which is 9.8 on Earth, and h is the height of the circle, which can be replaced with 2 times the radius of the circle, 2r.
Regarding forces, we know that the force in a circle is equal to the centripetal force, mv-squared over r. So at the top, we have the tension and gravity both contributing to this force, meaning that the tension at the top (T-top, measured in newtons) plus mg (the force of gravity) equals mv-squared over r.
But at the bottom, gravity is acting to reduce the centripetal force. So here the tension force minus mg is going to be equal to mv-squared over r. So, we now have an equation for the forces at the top and bottom. If you substitute one equation into the other, you find that the tension at the bottom is equal to the tension at the top plus 6mg.
Combined force equation
We can use these equations collectively to describe motion in a vertical circle and solve problems.
Practice problem 1: A ball on a string is whirled in a vertical circle. If the tension in the string at the top of the circle is 15 newtons and the ball weighs 0.1 kilograms, what is the tension in the string at the bottom of the circle?
First of all, we should write down what we know. The tension at the top (T-top) is 15 newtons and the mass of the ball (m) is 0.1 kilograms, and we want to find the tension at the bottom (T-bottom). To solve this, we can use this tension equation:
Diagram and equation for example
Plug the numbers in and solve for T-bottom, and we get 20.9 newtons. And that's our answer.
Over 79,000 lessons in all major subjects
Get access risk-free for 30 days,
just create an account.
Practice problem 2: A shopping bag can be treated more simply. If the shopping bag weighs 1 kilogram and the shopping bag is rotated in a vertical circle of radius 0.1 meters and the shopping bag is moving at 2.5 m/s at the bottom of the circle, how fast is it going at the top of the circle?
Writing down what we know, we see that mass (m) is 1 kilogram and radius (r) is 0.1 meters. We also know that the velocity at the bottom (v-bottom) is 2.5 m/s. We're asked to find the velocity at the top (v-top).
Here we're going to need the energy equation. We know everything in this equation except for v-top. So make v-top the subject, plug numbers in and solve, and we get 1.53 m/s.
Motion in a vertical circle is quite different to a horizontal one. This is because gravity is pointing towards the center of the circle at the top and away from the center of the circle at the bottom. This causes the speed to change (slowest at the top, fastest at the bottom) and means the tension also has to change during the motion.
Here are some equations we derived to describe this motion:
The energy equation says that one half mv-squared at the top (the kinetic energy) plus mg2r at the top (the gravitational potential energy) is equal to one half mv-squared at the bottom (the kinetic energy at the bottom). Where v-top is the velocity at the top measured in meters per second, v-bottom is the velocity at the bottom, also measured in meters per second, m is the mass of the object moving in a circle in kilograms, g is the acceleration due to gravity, which is 9.8 on Earth, and r is the radius of the circle, measured in meters.
We also have equations to describe the forces at the top and bottom, and an overall equation that tells us how the tension at the top relates to the tension at the bottom: that the tension at the bottom is equal to the tension at the top plus 6mg. We can use these equations collectively to describe motion in a vertical circle and solve problems.
After this lesson, you'll have the ability to:
Explain why speed and tension change during motion in a vertical circle
Identify the equations that describe motion in a vertical circle
Did you know… We have over 200 college
courses that prepare you to earn
credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the
first two years of college and save thousands off your degree. Anyone can earn
credit-by-exam regardless of age or education level.