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Today we have a beetle race. Beetle yellow and beetle red are stuck to this platform and this platform is rotating at a fixed speed.
Who’s faster – beetle red or beetle yellow. The answer to this question is at the end of this lesson. Before we find that
let’s look at some basic concepts of rotational motion.
See each of these rotating objects have a fixed axis about which the object is rotating
. Axis is fixed because it is at rest with respect to the surface of the earth.
All of them are rigid bodies-bodies the shape or size of the object does not change as they rotate.
So we’ve actually simplified things a little when we say that we are considering only rigid bodies and bodies that rotate about a fixed axis.
You can see that as the object rotates
each point on the object
is in circular motion with the centre on the axis.
Let me describe the motion of a point P, say.
This point moves from a to b to c to d.
To be a little more specific I can write down the x and y coordinates of a, b, c and d.
Let’s say a is at 3,2 and this is -1, 3.5 etc.
You have to tell me a way in which I can specify these positions on the circle using just one quantity.
If I use x and y coordinates I count this as two and I want to use just one quantity.
and I want to use just one quantity.
thought of something
Yes I think you have guessed correctly.
The angle that the line OP makes
with the x axis can give me the position of point P.
. If I say a is at 33.7 degrees you would immediately mark it here,
b is 106 degrees and d is -33.7 degrees.
now
I am specifying the angular position of a point on a circle
but I'm using just
one quantity
and that quantity is the the angle theta which the line OP makes with the X axis.
now you probably also noticed that this the angular coordinates can be either positive or negative
or negative. . The sign convention is the same as in trigonometry.
Here too angular displacements in the anticlockwise direction are positive and those measured in the clockwise direction are negative.
If you just recall motion along a straight line then you will remember that x in linear motion
is used to specify position whereas in rotational motion we are using the angle theta to specify angular position.
so theta gives me the angular position in rotational motion.
However there is another point that we need to take note of while describing rotational motion
I have used theta in degrees but normally in rotational motions all angles will be given in radians.
We will not use degrees.
The relation between degrees and radians is 360 degrees = 2 pi radians.
And I am sure you want to ask me why do we do this?
The reason for that is say I take a circle over here and this is an arc length s , this is the radius r and this angle is theta.
Now I can write theta = arc length/radius or s = r x theta.
See s is the arc length and theta is the angular displacement. This gives us a relation between the distance moved
along the circle and the angle covered at the centre of the circle.
Let’s put into a table all that we have learnt so far.
after displacement the next obvious quantity that we will look at is velocity
The speed v of the rotating particle
=(the distance covered)/t = (arc length,s)/time,t.
As the particle moves from P to Q it has made an angular displacement theta in time interval t.
Average angular velocity omega, w= (theta2 – theta 1)/(t2-t1).
w= (theta2 – theta 1)/(t2-t1).
If this is the x axis and the initial angle is theta 1 and
the final angular position at theta 2 then ( theta2 – theta1) gives me the angular displacement theta
This can be denoted as = delta theta/delta t.
now instead of the average value say I am looking at an instantaneous value
what do you mean by an instantaneous value?
I don’t want the velocity over a certain time interval but I want the velocity at every instant of time.
I want the velocity at every instant of time.
so now what I need is an instantaneous value
For an instantaneous value we take the limit as delta t very small
delta t approaches zero which means we consider an infinitesimally small time interval
in that case omega is equal to
omega = d(theta)/ dt.
see we are using calculus here but it is not difficult to understand
if I need the velocity at a certain instant of time then I have to make the time interval as small as possible
because if the time interval stays large I get an average value and not an instantaneos value.
Now if you take a look at our table you can see that
there should be a relation between the linear speed and the angular velocity
so now if you want to derive a relation between
linear velocity v and angular velocity w then how would you do that?
The hint is to look at the relation between linear displacement or the arc length and angular displacement.