A wheel of radius 14 inches is rotating 0.5 rads. What is the linear speed v, the angular speed

A wheel of radius 14 inches is rotating 0.5 rads. What is...

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Key Concepts

Conversion Between Radians per Second and RPM

Rotational speeds are often converted from radians per second to revolutions per minute (RPM) by first translating radians per second to revolutions per second (using the fact that 2? radians equal one revolution) and then multiplying by 60 to account for the number of seconds in a minute.

Relationship Between Angular Speed and Linear Speed

This concept explains how the linear speed (v) of a point on a rotating object is directly related to the product of the angular speed (?) and the radius (r) of the object, expressed as v = r?. It is crucial for understanding how rotational motion translates to linear motion.

Angular Unit Conversions

Often, angular speed or displacement may be given in different units such as radians, degrees, or revolutions per minute (RPM). Converting between these units involves understanding that 2? radians equal 360 degrees and one revolution, and using time conversion factors when shifting between units like seconds and minutes.

Converting Between Radians and Degrees

Angle measures can be converted from radians to degrees by multiplying by 180/?, and from degrees to radians by multiplying by ?/180. This conversion is essential when dealing with different systems of angular measurement within rotational motion problems.

Transcript

00:01 In this question, we are given that there is a wheel of radius 14 inches and it's rotating at 0 .5 radian per second.

00:06 This is radiant per second.

00:08 What's the linear speed? that is one.

00:10 So the linear speed is omega times the radius.

00:15 That's straightforward formula.

00:16 So omega is already given as 0 .5 radian per second and the radius is given as 14.

00:21 So that's got to be 7.

00:23 7 meter per second.

00:25 So that is the linear speed.

00:27 Then in part b they're talking about the angular rpm so that's where we need to do some calculations because we are given that the speed is 0 .5 radium per second.

00:39 Rpm is revolution per minute.

00:41 Now in 2 pi radiance in 2 pi radiance there is one revolution this is equal to the 1 revolution because one revolution has 2 pi degrees covered 2 by radiance covered.

00:54 So in 0 .5 radiance, in 0 .5 .5 radiance, the number of revolutions has got to be 1 over 2 pi times 0 .5 .5.

01:03 So that will be in revolutions per second because the angular speed is in 0 .5 radiance per second.

01:10 But so this is nothing but 0 .5 over 2 pi.

01:15 But we are interested in revolution per minute.

01:18 So if we have to convert the second to minute, then we have to multiply that about 1 over 16...

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