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Reduce the time interval

If we allow the time interval dT to be come shorter and shorter, we are averaging over smaller and smaller time intervals. In the limit, as dTapproaches zero, wAvg becomes w, which is the instantaneous angular velocity.

Angular velocity is a signed quantity

Angular velocity is also a signed quantity with the sign indicating the direction of rotation. By convention, counter clockwise rotation is viewed aspositive rotation. The sign of angular velocity is the same as the sign of the angular displacement that forms the basis for the angular velocity.

Units of angular velocity

The units of angular velocity are typically degrees per second or radians per second. You will learn later that radians is a dimensionless quantity.Therefore, when angular velocity is measured in radians per second, it often appears simply as

w = 10/sec

Radian measure

The most familiar measurement of angles, in the U.S. is in degrees. However, in some situations, it is more convenient to measure angles in radiansthan in degrees.

This becomes most apparent when we need to relate angular displacement or angular velocity with the distance traveled or the tangential speed of a point ona rotating object.

Definition of a radian

One radian is an angular measurement, which is equal to an angle at the center of a circle whose arc is equal in length to the radiusof the circle.

Simulate with a graph board

I recommend that you use your graph board to simulate an angle of one radian.

Using your graph board along with some string and pushpins, draw a Cartesian coordinate system. Then draw a circle with a convenient radius withits center at the origin of your coordinate system.

Make the arc match the radius

Cut a piece of string to the length of the radius of the circle. Then, beginning at the intersection of the circle and the horizontal axis, lay thestring along the circumference of the circle moving in a counter clockwise direction. Put a pushpin at the point where the string ends. Then stretch arubber band from that point back to the center of the circle.

Measure the angle

Using your protractor, measure the angle that the rubber band makes with the horizontal axis. That angle should be about 57.3 degrees, which is one radian.

Measure the number of radians in 360 degrees

Now, using the string whose length is equal to the radius of the circle as a measuring tool, determine how many strings of that length you can lay end-to-endaround the circumference of the circle.

You should find that about 6.28 (2*pi) such strings are required to go all the way around the circumference of thecircle.

An angle in radians is a ratio of lengths

An angle measured in radians is a ratio of two values, each of which have units of length. Therefore, such an angle has no dimensions.

Measurement of an angle in radians

angle = s/r


  • s is the length of an arc along the circumference of a circle
  • r is the radius of the circle
  • angle is the angle subtended by the arc length, s

An example measurement

Questions & Answers

what is the coefficient of -4×
Mehri Reply
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
I'm 13 and I understand it great
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
I rally confuse this number And equations too I need exactly help
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Commplementary angles
Idrissa Reply
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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Source:  OpenStax, Accessible physics concepts for blind students. OpenStax CNX. Oct 02, 2015 Download for free at https://legacy.cnx.org/content/col11294/1.36
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