When we study about motion, then in rotational dynamics, two important concepts come into mind: angular velocity and linear velocity. These terms and their relationship can be analyse the motion of objects in circular or rotational paths.
What is Angular Velocity?
Angular velocity (ω) describes the rate at which an object rotates about a fixed axis. It tells us how quickly an angle is swept out by a rotating object. It is a vector quantity, meaning it has both magnitude and direction.
The magnitude of angular velocity is given by:
ω = tθ where:
θ is the angular displacement (measured in radians),
t is the time taken.
The SI unit of angular velocity is radians per second (rad/s).
What is Linear Velocity?
Linear velocity (v) is the rate at which an object moves along a path. It measures how quickly an object covers distance and is given by: v = ts where:
s is the linear displacement (distance covered),
t is the time taken.
The SI unit of linear velocity is meters per second (m/s).
Relationship between Angular and Linear Velocity
The relationship between angular velocity (ω) and linear velocity (v) becomes clear when we consider an object moving in a circular path. In such cases, the linear velocity is related to the radius (r) of the circular path and the angular velocity:
v = r⋅ω Here:
r is the radius of the circular path,
ω is the angular velocity.
This equation tells us that the linear velocity of an object is directly proportional to both the angular velocity and the radius of the circular path.
Relationship
1.Proportionality: If the angular velocity is constant, a larger radius results in a greater linear velocity. Like, on a spinning disc, a point near the edge moves faster (has higher linear velocity) than a point closer to the center, even though both points have the same angular velocity
2. Direction:
The linear velocity at any point on a circular path is always tangential to the circle, means it points in the direction of motion along the edge of the circle.
The angular velocity vector points along the axis of rotation, follows the right-hand rule.
3. Practical Examples:
Wheels and Gears: When a wheel rotates, its angular velocity determines the linear velocity of its rim. This principle is vital in designing vehicles and machinery.
Earth’s Rotation: The Earth rotates with a specific angular velocity, but the linear velocity of a point on the equator is much higher than that of a point near the poles due to the larger radius at the equator.
Key Points
Both angular and linear velocity are essential for describing motion in circular and rotational systems.
The equation v = r⋅ω serves as a bridge between these two types of velocities.
Angular velocity is uniform for all points on a rotating object, linear velocity varies depending on the radius.
In pure rotational motion, all parts of the body move with the same angular velocity at any instant of time.
For rotation about a fixed axis, the angular velocity vector lies along the axis of rotation . Also, it points out in the direction in which a right handed screw would advance, if the head of the screw is rotated with the body.
The relation between linear velocity vector v and angular velocity vector ω is given by v = ω . r
Angular acceleration:- It is the time rate of change of angular velocity. α = d ω / dt
Note
Angular velocity and linear velocity are closely related but describe different aspects of motion. Angular velocity focuses on the rotational rate, while linear velocity deals with the tangential speed of a point on the rotating object.
Their relationship allows us to solve problems in rotational dynamics and analyse, phenomena like planetary motion, vehicle design, and amusement park rides.
Angular velocity (ω) is a measure of how quickly an object rotates or revolves around a fixed axis. It indicates the rate of change of angular displacement and is measured in radians per second (rad/s).
Linear velocity (v) is directly proportional to the angular velocity (ω) and the radius (r) of the circular path. The relationship is given by the formula: v = r⋅ω
This means that for a larger radius or higher angular velocity, the linear velocity increases.
Angular velocity describes how fast an object rotates (rate of change of angle).
Linear velocity describes how fast an object moves along a straight or curved path (rate of change of distance).
Angular velocity is measured in radians per second, while linear velocity is measured in meters per second.
Yes, objects on the same rotating body can have the same angular velocity but different linear velocities depending on their distance from the axis of rotation. Points farther from the center have greater linear velocity due to the larger radius.
In circular motion, the direction of linear velocity is always tangential to the circle at any given point. It changes continuously as the object moves along the circular path.
If the radius (r) is doubled while keeping the angular velocity (ω) constant, the linear velocity (v) also doubles. This is because v is directly proportional to r according to v = r⋅ω
Angular and linear velocity are encountered in various scenarios such as:
Wheels of vehicles (rotational motion generating linear motion).
Gears and pulleys in machinery.
Circular motion of celestial bodies like planets.
Amusement park rides with rotating parts.