Angular Kinematics
![[url=https://pixabay.com/en/night-stars-rotation-starry-sky-1846734/]"Polaris"[/url] by Pexels is in the [url=http://creativecommons.org/publicdomain/zero/1.0/]Public Domain, CC0[/url]
A photo taken toward the north star Polaris. The rotation of the earth is evident due to the apparent motion of the stars. The trails they leave all subtend the same angle. Stars travel along paths proportional to distance from the center of the rotation.](https://www.geogebra.org/resource/XGWcnbzq/jmkJGRDQaDOdfdfY/material-XGWcnbzq.png)
Rotation versus Translation
When objects accelerate in the sense that we can describe it in terms of a vector and relate that to a velocity and a displacement, it is called translational motion. In other words, nearly everything we've discussed this semester so far has dealt with translational motion.
Now we turn to rotational motion. As the name suggests, this motion implies not that the object is going anywhere, but that it is spinning. For a rigid body (one which does not have movable parts), these are the only two types of motion it can undergo. In other words, every ball that flies or wheel that rolls or rock that gets tossed can be described by a combination of translational and rotational motion. The focus in this chapter is the rotational part of that discussion.
Review of Angular Variables
Recall that given a circular arc segment of radius r that subtends an angle , that the arc length is Taking a time derivative of both sides gives us our definition of angular velocity: The (omega) is NOT a w (double u)! Note that represents the tangential velocity.
Taking one more time derivative gives us an expression for angular acceleration: The tangential acceleration is
So in short:
EXAMPLE: Suppose a CD is rotating at 500rpm and has a diameter of 12cm. What is the speed of a point on the edge of the CD?
SOLUTION:
EXAMPLE: How fast is a person traveling with respect to earth's center while standing on the surface at latitude ?
SOLUTION: Recall that the radius of the circular motion is earth's radius times the cosine of the latitude, and that must be expressed in radians per second. The earth rotates at roughly The earth's radius is Putting this together gives
A Radian is a Ratio
Notice that the units on these expressions can look odd. The radians on the right side terms just vanishes. This is ok since a radian is not a unit, but a ratio of arc length to radius in a circle. Some people prefer to call it a unit, but in any case a radian is dimensionless. This disappearance of units should not happen with other units that you encounter, but it happens all the time when converting from translational terms to rotational terms.
Angular Kinematic Equations
If you take what you know of ordinary kinematics and apply it to angular kinematics, all of it still holds. Recall that our old definitions were:
EXAMPLE: A merry-go-round starts from rest and accelerates at (assuming the z-axis is the upward direction) until reaching an angular velocity of How long did this take, how many times around did the children travel while this was taking place, and what was the total angular displacement (vector)?
SOLUTION: Using This means and we set that equal to find the time it took to get up to speed. It is To find the angular displacement, use gives or 1.8 rotations.
EXAMPLE: A child sits 1.5m from the axis of rotation of the merry-go-round from the previous example. What is the child's speed as a function of time? What is the child's total acceleration as a vector? Please use and to describe the vector. Assume the polar coordinates are in the x-y plane.
SOLUTION: We use to find the speed, which only has a component tangent to the path. This gives The total acceleration vector is made of two orthogonal components - the tangential acceleration in the direction and the inward centripetal acceleration. Recall that the centripetal acceleration may be written and that as a vector we may write Put together this gives the total acceleration of the child: