Askiitians Tutor Team
Last Activity: 4 Months ago
To derive the relation between linear velocity (vv) and angular velocity (ω\omega), we need to understand the connection between the motion of a rotating object and the linear motion of a point on that object.
Key Concepts:
• Angular velocity (ω\omega): This is the rate at which an object rotates around a fixed axis. It is measured in radians per second (rad/s).
• Linear velocity (vv): This is the velocity of a point on a rotating object. It is measured in meters per second (m/s).
• Radius (rr): The distance from the center of the rotation (axis of rotation) to the point on the object whose velocity is being considered.
Step 1: Define Angular Displacement and Linear Displacement
• Angular displacement (θ\theta) is the angle through which a point or line has been rotated about a specific axis.
o θ\theta is measured in radians.
• Linear displacement of a point on the object is the distance traveled along its circular path, which is related to the angular displacement by the radius rr.
The arc length ss (linear displacement) traveled by a point on the circumference of a circle is given by:
s=rθs = r \theta
where rr is the radius of the circle and θ\theta is the angular displacement in radians.
Step 2: Angular Velocity and Linear Velocity
• Angular velocity (ω\omega) is the rate of change of angular displacement with respect to time:
ω=dθdt\omega = \frac{d\theta}{dt}
where ω\omega is in radians per second.
• Linear velocity (vv) is the rate of change of linear displacement with respect to time:
v=dsdtv = \frac{ds}{dt}
Step 3: Relating Linear and Angular Velocity
From the equation for linear displacement s=rθs = r\theta, we can differentiate both sides with respect to time to find the relationship between linear velocity and angular velocity:
dsdt=ddt(rθ)\frac{ds}{dt} = \frac{d}{dt}(r\theta)
Since rr is constant, this simplifies to:
v=r⋅dθdtv = r \cdot \frac{d\theta}{dt}
But dθdt\frac{d\theta}{dt} is the definition of angular velocity ω\omega, so:
v=r⋅ωv = r \cdot \omega
Final Relation:
Thus, the linear velocity vv is related to the angular velocity ω\omega by:
v=r⋅ωv = r \cdot \omega
Conclusion:
• The linear velocity vv of a point on a rotating object is equal to the product of the radius rr (distance from the axis of rotation) and the angular velocity ω\omega.
• This relationship shows that as the angular velocity increases, the linear velocity also increases, and the linear velocity is directly proportional to the radius of rotation.