How to Calculate Tangential Velocity

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you'll learn what information you need to calculate your tangential velocity. You'll also learn the easy formula that tells you just what to do with your values.

The Steps

Did you know that when you ride your bicycle, your bicycle actually is traveling at two different speeds? One speed is your tangential velocity, your speed going forward and the other is your angular velocity, the speed at which your wheel is turning. They're different because one measures how fast your wheel is spinning (angular velocity) while the other measures the speed at which the wheel is moving forward (tangential velocity).

tangential velocity

Because your wheel is round, the angular velocity and tangential velocity are related by a single variable with a very simple formula.

Let's take a look.

Find the tangential velocity of a bicycle whose wheels have an angular velocity of 10 pi radians per second and a radius of 12 inches. Angular velocity is given in units of radians per a time unit such as seconds. When a wheel has made one whole turn, it will have turned 2 pi radians. So a wheel with an angular velocity of 10 pi radians per second means that the wheel turns completely 5 times each second. To convert this angular velocity to tangential velocity, you would follow these steps.

Step 1: Recall the formula that shows the relationship between tangential velocity and angular velocity.

The very first step has you recalling the formula that shows you the relationship between these two velocities.

tangential velocity

This formula tells you that your angular velocity (represented by the Greek letter omega) times the radius (r) is equal to your tangential velocity (vt).

Step 2: Plug in your angular velocity for the Greek letter omega and your radius in for r.

For these types of problems, you are given the angular velocity. You take this angular velocity and plug this into your formula. Make sure your units are in radians per a time unit. Your time unit can be either seconds, minutes, or hours. Then you plug in your radius value for r.

For your problem, you plug in 10 pi radians per second for the Greek letter omega and 12 inches for your radius r.

tangential velocity

Step 3: Evaluate your formula to find your tangential velocity.

Last, you evaluate to find your answer, multiplying where necessary.

For your problem, you multiply your 10 pi by 12.

tangential velocity

You get 376.99 inches per second. You can leave your answer like this or you may need to change the units into something else such as miles per hour.

To change your units, you use your unit conversion values such as 12 inches in one foot and 5,280 feet in a mile. Making the conversions, you get this.

tangential velocity

The Solution

Your solution here is 376.99 inches per second or 21.4 miles per hour.

This tells you that if you are able to make your bicycle wheel spin 5 complete turns each second, then you'll be traveling at a speed of 21.4 miles per hour.

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