M2.2 Uniform Circular Motion
Circular motion is a motion of an object along the circular path. Our object will be considered as a point or at least very small object, so its dimensions can be neglected.
To analyze this motion we must define displacement, velocity and acceleration for this particular type of motion. We will do it with the help of Fig. M2.4
Change of position in a particular direction. During circular motion the direction changes continuously and the displacement must be defined for a very short time of motion. Such “very small” value is called in calculus infinitesimally small and enters equations as the name of the quantity proceeded by letter d.
In Fig. M2.5 there is a small part of an arc drawn during circular motion by a moving object. The displacement defined as a change of position from 1 to 2 does not coincide with the path of a moving object as long as the distance from 1 to 2 is greater then 0. Taking position 2 closer and closer to position 1, which we can write in the form
we end up with an infinitesimally small displacement
which can be considered as coinciding with the path of the moving object.
In this motion we distinguish two velocities – linear and angular.
Linear velocity is defined just like for motion along a straight line
and depicted in Fig. M2.4 at 4 different points of a circular path.
The direction of velocity is different at each point of a path, but its magnitude is constant in uniform circular motion. “Uniform” denotes motion with velocity of constant magnitude, but the above definition of instantaneous velocity is valid for nonuniform circular motion also. As a matter of fact it is valid for any type of motion.
The direction of this velocity is always perpendicular to the radius of a circle drawn to the point at which velocity is calculated so it is tangent to the circle at that point. Therefore this velocity is also called tangential velocity.
Angular velocity usually denoted ω is defined as the time rate of change of angular displacement.
This angular displacement is an angle α a between two consecutive positions of a radius in Fig. M2.5. This angle is usually expressed in radians.
Radian:by definition it is the angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle. For example
3600 = 2π radians (SI notation rad)
900 = 2π/4 rad = π/2 rad
2π = 6.283185 (approximately)
1 rad = 360/(2π)o = (180/π)o (approximately 57.29578o).
So the angular velocity can be written in the form
Eq. M2.18 defines instantaneous angular velocity so it is valid for any type of circular motion, not only uniform, that is with constant angular velocity. We call it velocity, so it must be a vector.
How can the angle be a vector? What is the direction of this vector?
We will explain this step by step. First, we introduce angular speed, the quantity analogous to linear speed or simply speed, as we called it in the “Straight line motion” Chapter. Equation M2.19 is the definition of angular speed also called rotational frequency or angular frequency.
To “change” this speed to velocity a direction must be assigned to it.
The direction of the angular velocity vector is perpendicular to the plane in which the rotation takes place. If the rotation appears counterclockwise with respect to an observer, then the vector of angular velocity points towards the observer. The direction of angular velocity vector can be also derived from the right hand rule which was applied to define the z axis direction in a Cartesian coordinate system.
We now introduce two other quantities used to describe uniform circular motion.
Period T - the time it takes the object to make one revolution, units – second ( s ).
Frequency f (or ν), unit - s-1.
f = 1 / T (M2.20)
For the uniform circular motion the angular speed is constant, and we can write
2π is a 360o angle expressed in radians and T is a period, and after substituting T from Eq. M2.20
Relation between different parameters defined for uniform circular motion.
Relations are given for scalar quantities to avoid vector notation, which not everyone is familiar with.
2πr, where 2π is an 360o angle expressed in radians, is simply the length of circumference of a circle with radius r. So the first part of Eq. M2.23 states that the total length of the path traveled by an object divided by the time required to travel it, is equal to the linear speed in this motion.
Acceleration in uniform circular motion.
It was mentioned that velocity has constant magnitude, but changes direction continuously. Formally, if the velocity is changing there must be acceleration. In the uniform circular motion it is
Centripetal (radial) acceleration
which is directed towards the center of the circle on which the object travels.
In Fig. M2.6 there is a schematic drawing which is used to derive the formula for centripetal acceleration.
The velocity change Dv is a vector directed towards the center of a circle. The inset shows vectors representing velocities at two instants of time, with their origins moved to one point. Velocities at any point are perpendicular to the radius at this point, so the triangles
Δv / v = Δr / r (M2.24)
For very small values of Δv and Δr and for angle expressed in radians there is a relation
Δr = r Δα (M2.25)
We divide both side of Eq. M2.24 by Δt and substitute Δr / r evaluated from Eq. M2.25 to get
Taking the limit as that is
dv/dt is simply acceleration and
dα/dt is angular speed ω.
We must remember that the change of the velocity v which is included in the above equations is along the radius r and directed towards the center of a circular path of a moving object. Because of that we this acceleration is called centripetal or radial. Denoting this acceleration by ac we have
or, after multiplying both side by v
From Eq. M2.23 we have
ω = v / r
and substituting this into Eq.M2.28
ac = v2 / r (M2.29)
With Eqs. M2.20 to Eq. M2.23 we can have few equivalent equations for centripetal acceleration
Remember, all these equations are for uniform circular motion, that is, motion with constant angular velocity. Some problems attached to this chapter allows you to master this type of motion.