PHY C7: Kinematics of Uniform Circular Motion

Today we’ll be covering:

  • Angle Conventions
  • Angular displacement
  • Angular speed
  • Formula: v = rω

Let’s go!


Angles
When describing motion of an object following a circular path,
angles (θ) are more useful when measured in RADIANS, not degrees.

For SMALL values of angles (< 5 degrees, < 0.0873 radians), you can make these approximations:

  • sin x = x
  • tan x = x

Ex:

  • sin 0.06 = 0.06
  • tan 0.02 = 0.02
  • But sin 1.5 ≠ 1.5

What is Angular Displacement?
Angle moved through by an object.
It is denoted as ∆θ.

What is Angular Speed?
Rate of change of angular displacement.
It is denoted by the symbol ω (omega).

Angular Speed = Change in angle over time
ω = ∆θ/t

In units,
rad s-1

What is Angular Velocity?
Angular speed in a given direction.

Speed is a scalar, but velocity is a vector.
Thus, the angular speed of an object can be constant, but its angular velocity changes depending on direction.

The magnitude of angular velocity = angular speed.
Thus, both have the same units: rad s-1


Calculating Angular Speed given Period
The period (T) of an object’s rotation is the time taken to complete 1 rotation.

Since a complete circle has 2π rad, & it completes that angular displacement in T seconds,
ω = 2π/T

Calculating Angular Speed given Frequency
Sometimes the frequency of an object’s rotation is given as RPM (rotations per minute) or RPS (rotations per second).

To find angular speed,

  1. Find the frequency (f) in Hertz (Hz) of the rotation
  • 1 RPS = 1 Hz
  • 1 RPM = 1/60 RPS = 1/60 Hz
  1. Use ω = 2π/T
    Since T = 1/f,
    ω = 2πf

What is Linear Speed?
Rate of change of distance as an object travels following the circumference of a circle.

You should know that:
the length of an arc subtended by angle θ = radius x angle θ
s = rθ
∆s = r∆θ

Dividing both sides by time to find the RATE of change of distance:
∆s/∆t = r∆θ/∆t
v = r∆θ/∆t

Since ω = ∆θ/∆t,
v = rω

In units,
ms-1

Two objects travelling in a concentric circle can have the SAME angular speed, but DIFFERENT linear speed, IF they travel at different RADII from the centre of the circle.

The object with a larger radius actually travels at a HIGHER linear speed than the object closer to the centre, if both have the same angular speed.


What is Linear Velocity?
Linear speed in a given direction.

Speed is a scalar, but velocity is a vector.
As with angular speed vs angular velocity,
the linear speed of an object can be constant, but its linear velocity changes depending on direction.

The magnitude of linear velocity = linear speed.
Thus, both have the same units: ms-1


⇐ Previous in Physics: Power
⇒ Next in Physics: Centripetal Acceleration & Force

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