Question #c4cde

1 Answer
Mar 31, 2015

This is a very good question even if I think it is quite difficult and it will need an equally good answer; probably my answer is not good and is not going to be completely satisfactory and explanatory, but anyway;

Rotational kinematic is quite similar to normal linear kinematic where you deal with movement along lines (for example, along the x-axis).
The big difference is that instead of using a linear distance, say, #d# you use angles such as #theta# to describe the movement along a circle of radius #r#.

Every concept in rotational motion is equivalent to its linear counterpart BUT it must involve the angle and the radius:
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So the distance from A to B given by #d# on a circle turns to be an ARC of length #s# given as:
#s=r*theta# where the angle #theta# is in RADIANS.

Next consider another kinematic quantity, velocity. In linear terms you have that if the motion is uniform (no acceleration) your velocity is simply:
#v=(x_B-x_A)/(t_B-t_A)#
you can imagine that in rotational motion angles have to get into it so you have a "kind" of velocity called Angular Velocity , #omega#:
#omega=(theta_B-theta_A)/(t_B-t_A)#.

enter image source here

The good thing is that we can "bridge" from circular to linear by using a key...the radius!
So you have: #v=omega*r#

You may start to get used now to this parallel between the two worlds and imagine that Angular Acceleration , #alpha#, will be the change in velocity (angular) with time:

#alpha=(omega_B-omega_A)/(t_B-t_A)#

The connection between rotational and linear is a little bit more complex for the acceleration.

You have that #alpha# in the linear world is known as Tangential Acceleration, #a_(tan)#, with #a_(tan)=r*alpha#. Remember that this is one of the possible components of your total acceleration, the other being the Centripetal Acceleration given as #a_c=v^2/r=omega*r# (try to substitute for #v# and check this result by yourself!).

The formulae for the linear and rotational kinematics have the same form in which instead of distance you have angles:
A. Beiser, Scaum's Outline - Applied Physics 3rd ed., McGraw-Hill, 1994

You use them exactly as in the linear world but you must remember that when you go back to linear you have to rearrange them by using the radius of your circular trajectory (as in #v=omega*r#)!

Ok...hope it is not completely confusing...!