2D Circular Motion

Interactive Lecture

Understanding Angular Motion and Centripetal Forces

by Dr Arnaldo Delli Carri

Fundamental Concepts

Angular Velocity

Angular velocity (\(\omega\)) represents the rate of change of angular displacement. It describes how quickly an object rotates about a fixed axis.

Angular Velocity: \[\omega = \frac{d\theta}{dt} = \frac{v}{r}\] where: \(\omega\) = angular velocity (rad/s), \(\theta\) = angular displacement (rad), \(v\) = linear velocity (m/s), \(r\) = radius (m)

Uniform Circular Motion and Centripetal Acceleration

When an object moves in a circle at constant speed, it experiences centripetal acceleration directed towards the centre. Despite constant speed, the velocity vector continuously changes direction.

Centripetal Acceleration: \[a_c = \frac{v^2}{r} = \omega^2 r\] Centripetal Force: \[F_c = ma_c = \frac{mv^2}{r} = m\omega^2 r\]

For a bead whirled on a string, the string tension \(T\) is the centripetal force: \(T = F_c = m\omega^2 r\).

In uniform circular motion, the acceleration is always perpendicular to the velocity, causing only a change in direction, not speed.

Angular Kinematics - SUVAT Equivalents

The equations of angular motion parallel those of linear motion, with angular quantities replacing linear ones:

Linear Motion Angular Motion Relationship
\(v = u + at\) \(\omega = \omega_0 + \alpha t\) \(v = r\omega\)
\(s = ut + \frac{1}{2}at^2\) \(\theta = \omega_0 t + \frac{1}{2}\alpha t^2\) \(s = r\theta\)
\(v^2 = u^2 + 2as\) \(\omega^2 = \omega_0^2 + 2\alpha\theta\) \(a_t = r\alpha\)
\(s = \frac{1}{2}(u + v)t\) \(\theta = \frac{1}{2}(\omega_0 + \omega)t\) -

Angular Acceleration

Angular acceleration (\(\alpha\)) is the rate of change of angular velocity. It describes how quickly the rotation speed changes.

Angular Acceleration: \[\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}\] where: \(\alpha\) = angular acceleration (rad/s²)

Accelerated Circular Motion and Tangential Acceleration

When angular velocity changes, the object experiences both centripetal and tangential accelerations:

Tangential Acceleration: \[a_t = r\alpha\] Total Acceleration: \[a_{total} = \sqrt{a_c^2 + a_t^2}\] where: \(a_t\) = tangential acceleration (m/s²)
In non-uniform circular motion, the total acceleration has two components:

Interactive Simulation: Bead on a String

System Properties
ω = 2.00 rad/s
α = 0.00 rad/s²
v = 3.00 m/s
ac = 6.00 m/s²
at = 0.00 m/s²
atotal = 6.00 m/s²
Tension:
T = 6.00 N
Simulation Legend
● String Tension
● Velocity Vector
● Centripetal Acc
● Tangential Acc
● Total Acc
2.0 rad/s
0.0 rad/s²
150 px

Engineering Applications

Circular motion governs the behaviour of these engineering systems:

Summary of Key Equations

Concept Equation Description
Linear-Angular Relationship \(v = r\omega\) Linear speed from angular velocity
Centripetal Acceleration \(a_c = \frac{v^2}{r} = \omega^2 r\) Acceleration towards centre of circle
Centripetal Force \(F_c = \frac{mv^2}{r} = m\omega^2 r\) Force required for circular motion
Tangential Acceleration \(a_t = r\alpha\) Tangential component (non-uniform motion)
Total Acceleration \(a_{total} = \sqrt{a_c^2 + a_t^2}\) Magnitude of resultant acceleration
Angular Kinematics \(\omega = \omega_0 + \alpha t\)
\(\theta = \omega_0 t + \frac{1}{2}\alpha t^2\)
\(\omega^2 = \omega_0^2 + 2\alpha\theta\)		 Equations for angular motion (analogous to linear)

Practice Problems

Problem 1: Uniform Circular Motion

A bead of mass 50 g is attached to a string of length 0.8 m and rotates in a horizontal circle at a constant angular velocity of 5 rad/s. Calculate:

  1. The linear speed of the bead
  2. The centripetal acceleration
  3. The tension in the string

Show Solution

Problem 2: Angular Acceleration

A wheel starts from rest and accelerates uniformly to 120 rpm in 4 seconds. Find:

  1. The angular acceleration
  2. The number of revolutions made in these 4 seconds
  3. The tangential acceleration of a point 0.3 m from the axis

Show Solution

Problem 3: Non-Uniform Circular Motion

A particle moves in a circle of radius 2 m with an angular velocity that increases from 2 rad/s to 6 rad/s over a distance of \(4\pi\) radians. Calculate:

  1. The angular acceleration
  2. The time taken for this change
  3. The magnitude of total acceleration when \(\omega = 4\) rad/s

Show Solution

Knowledge Check Quiz

Question 1: In uniform circular motion, which statement is correct?

Question 2: A particle moves in a circle of radius 0.5 m with angular velocity 4 rad/s. What is its centripetal acceleration?

Question 3: When a string is cut during circular motion, the object:

Question 4: The relationship between angular acceleration \(\alpha\) and tangential acceleration \(a_t\) is:

Question 5: A wheel completes 10 revolutions in 5 seconds starting from rest with uniform acceleration. What is the angular acceleration?