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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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• 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.
• 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.
• 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 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.
• I have a question about how to solve a problem like this:

An object with an initial velocity of 0.12 rad/s (0.12 radians per second) accelerates at 0.11 rad/s^2 (0.11 radians per second squared) over a distance of 0.25 radians. What is the final angular velocity (ω) of the object in r/s (radians per second)?

I know that to solve for angular velocity, you can use this equation:

ω = ωₒ + α * t

where ωₒ = initial velocity, α = angular acceleration, and t = time.

However, I was not able to substitute everything needed to solve for ω:

I wasn't given a time. Originally I figured, "Oh, well no big deal. I can probably just do something with the numbers to solve for time, or maybe look through all my equations and then solve one for t..."

I was wrong (at least to my knowledge).

I had no equations where I could solve for t, because I had multiple unknowns...

Eventually I started using intuition rather than solid numbers & equations, and I tried thinking about graphs. I figured that if the initial velocity was 0.12 rad/s, t at that point would equal 0, and that before the origin it was a constant line (with a y coordinate of 0.12). Once it got past 0, however, there would be an (angular) acceleration of 0.11 rad/s^2, so it would be a parabola shape.

It helped me to think of it this way (ignore the part where it's a really, really fast car and this car would have destroyed itself quite quickly):

You have a car going at a constant velocity of 12 km/s (ωₒ), and once you pass a white line across the ground, t is now > 0 and you start accelerating 11 km/s (α) until you have driven exactly 25 km (x) past the white line. I figured that if you started at 12 km/s (ωₒ) and started accelerating for 25 km, at roughly the same speed, it's kind of like doubling it, right? You were adding 11 km/s every second, and so after 1 second, you were going 23 km/s, and so t would have had to have been just over one second to reach the 25 km mark. Of course, I didn't know exactly how much time had elapsed; I just knew it was a bit more than a second.

So I just started entering numbers into the computer, starting at 0.24, going up one hundredth each time. I had only gotten to 0.25 when it said I was correct.

"Aha! I... guessed the right answer, how the heck are you supposed to calculate t without guessing?"

I put 0.25 in my original angular velocity equation:

and then solved for t this way:

t ≅1.2 seconds

I'm glad I finished the problem after agonizing over it for hours, but I am still not satisfied with the issue: How are you supposed to calculate t when given initial velocity, angular acceleration, and distance? Maybe I am just missing an equation from my equations sheet, but this was (and is) a very frustrating problem... help is warmly welcomed.
• i suggest you to use the eqn
(final angular velocity)^2=(initial angular velocity)^2+2(angular acceleration)(angular displacement)
(1 vote)
• What is the difference between Δ𝚹 and |Δ𝚹| ?
(1 vote)
• |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?
(1 vote)
• 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.
(1 vote)
• How do we find constant velocity if we do not have the radius, but we have change in time, acceleration, and angular displacement?
(1 vote)
• i suggest you watch the "rotational kinematic formula" video in the APphysics 1( torque and angular momentum)play list
(1 vote)
• how to find angular velocity about a point on the circumference? how is it different from angular velocity about center of rotation?
(1 vote)
• 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.
(1 vote)