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# Angular velocity and speed

In uniform circular motion, angular velocity (𝒘) is a vector quantity and is equal to the angular displacement (Δ𝚹, a vector quantity) divided by the change in time (Δ𝐭). Speed is equal to the arc length traveled (S) divided by the change in time (Δ𝐭), which is also equal to |𝒘|R. And arc length (S) is equal to the absolute value of the angular displacement (|Δ𝚹|) times the radius (R).

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• So at , Sal expresses the speed in meters/second, when he expresses the angular velocity in radians/second. The only thing that was different about the speed and angular velocity was that the magnitude of the angular velocity was multiplied by a scalar with the "meters" unit. Therefore, following algebraic logic, the unit for speed should be meters*radians/second.

Where did the "radian" part go? Does it not count as an actual unit? And, if it doesn't, couldn't you replace the "radian" part with 1 so that you would have 1/second=1 Hertz? Hertz would be a much easier unit to use in cases like these.

Please tell me whether or not I am just confusing myself for no reason. Thanks!
• pi radian = R pi meter. Both radian and meter are units for length. Radian is only a more convinient way of describing circular distance.
In this case, pi radian=7pi. By multiplying pi/2 by 7 (R=7), he automatically converts the unit from radian to meter.
• Are speed and linear velocity the same?
• Kind of. Speed is a scalar quantity and velocity is a vector quantity so it has direction and magnitude.
• Can't we also use the Speed=Distance/Time formula where the distance is 1/4th the circumference of the circle to find out the speed?
• Absolutely! The arclength formula is actually derived from multiplying the portion of the circle considered (in this case, 1/2pi rad out of 2pi rad, or 90' out of 360') by 2pi(radius). From this formula (2piR times portion of circle), you get a simplified formula for arclength: S = r|delta theta| For a detailed explanation, see arclength from angular displacement video.

Anyway, finding speed in this video, you can use that arclength formula divided by time to find distance travelled over time -- speed.

Note that r(delta theta)/t equals r multiplied by (delta theta)/t, which is the same thing as angular velocity (w). So we conclude that v = wr

To state clearly the answer to your question, you absolutely do use distance/time by the circumference of the circle in the way arclength is calculated! Sal just draws another relationship between angular velocity and speed.
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• what i understand is that for calculating Angular velocity we take the difference of angles and divide it over the time.but what if the ball moves back and forth.in this case speed would be different from Angular velocity.what i'm missing?
• Remember that velocity is defined as the change of position over the change of time.

To find angular velocity for a ball moving back and forth, you will have to find the beginning angle and the final angle at where the ball stops moving. Then find the difference between the two angles and divide by time.

To find the speed for a ball moving back and forth, you will have to find the total distance the ball moved. And don't forget to avoid a common misconception. For example, if the ball moved 90 degrees and then -90 degrees, the total distance is NOT ZERO. The displacement equals zero, but not the total distance.
• Since this is the average angular velocity how do we calculate the instantaneous angular velocity?
• if the object has a constant angular acceleration, the instantaneous angular velocity at any time "t" is just the initial angular velocity plus(angular acceleration*t). sorry or bad writing though
• What is the difference between Δ𝚹 and |Δ𝚹| ?
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• |Δ𝚹| means no matter what the sign of Δ𝚹 is, the amount is positive, as indicated by | |.
• Let's say a ball with a radius of .11 m is rolling at 5m/s. To find angular velociy, you multiply by the radius?? Why? What are the new units?
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• well, speed=abs(angular velocity)*r. and if you want to find the angular velocity. you would just divide the speed by the radius.the direction depends on the fact of being clockwise or not. if its clockwise, its negative and if its counter clockwise its positive.and the units are radians/sec.and its because that's how we derived the formula.
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• How do we find constant velocity if we do not have the radius, but we have change in time, acceleration, and angular displacement?
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• i suggest you watch the "rotational kinematic formula" video in the APphysics 1( torque and angular momentum)play list
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• how to find angular velocity about a point on the circumference? how is it different from angular velocity about center of rotation?
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• I think you're talking about the ball's velocity around the circumference, right?
If yes, then it's the same thing as finding the speed in the "Angular velocity and speed" video and putting a "+" or "-" to indicate direction.
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• When the initial calculation was done for angular velocity was this just for 1 second? I am asking because that calculation included time and then Sal multiplied another 3 seconds to get something else (or I have no clue what because I can't read minds).
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