3. Rotational motion
Rotational motion (or we can say circular motion) can be analyzed in the same way of linear motion. In this section, we will find the velocity, acceleration and other concepts related to the circular motion. Uniform circular motion is one of the example. In uniform circular motion speed of the object is always constant and direcrotational motion imagetion is changing. Thus, velocity of the object is changing and as a result object has acceleration. Some concepts will be covered like: rotational speed (angular speed), tangential speed (linear speed), frequency, period, rotational inertia of the objects, torque, angular momentum and its conservation.
A particle moves in a circle of radius r. Having moved an arc length s, its angular position is θ relative to its original position, where
.
In mathematics and physics it is conventional to use the natural unit radians rather than degrees or revolutions. Units are converted as follows (proportion rules)
An angular displacement is a change in angular position, like
,
where is the angular displacement,
is the initial angular position and
is the final angular position.
Linear speed and tangential speed gives the same meaning for circular motion. It can be defined as distance taken in a given time. If the object has one complete revolution then distance traveled becomes , which is the circumference of the circle object. In this terms we get
.
Time passing for one revolution is called period. The unit of period is second. T is the representation of period.
Number of revolutions per one second. The unit of frequency is 1/second. We show frequency with letter f. Unit for this quantity is 1 Herz = 1/s = 1 Hz.
Then the equation of tangential speed becomes
.
We define angular velocity as “change of the angular displacement in a unit of time”. One total rotation corresponds to . radians. Units of angular speed are revolution per unit time radians per second. We show angular speed with the Greek letter
. All points on the platform have same angular velocity.
.
.
Direction of the speed changes as time passes and always tangent to the circle. Change in the direction of velocity means system has acceleration which is called angular acceleration. Because of the direction of acceleration, we call it centripetal acceleration.
Mathematical representation of centripetal acceleration is
,
- sign in front of the formula shows the direction with respect to the R position vector. We can rewrite centripetal acceleration in terms of angular velocity and tangential velocity. The formula could be also write as
or
.
If there is acceleration then we can say there must be also a net force causing that acceleration. The direction of this net force is same as the direction of acceleration which is towards to the center. From the Newton’s Second Law of Motion we get that
and it could be referred as
,
where
m is mass of the object, r is the radius of the circle, T is a period and V. It could be also revritte as
.
When a car goes around a curve we feel that as if something pulls us outward to the center of that curve. In real, of course there is no such a force exerting on us. Because of the Newton’s First Law of Motion “Law of Inertia”, we feel that something pulls us outward. This is only inertia of us, in real there is no centrifugal force.
We define torque as the capability of rotating objects around a fixed axis. In other words, it is the multiplication of force and the shortest distance between application point of force and the fixed axis. From the definition, you can also infer that, torque is a vector quantity both having direction and magnitude. Torque is shown in physics with the symbol . You can come across torque with other name “moment of force”.