The relationship between angular and linear velocities describes how a point on a rigid body rotating about a fixed axis moves. If a body rotates with an angular velocity 𝜔, a point at a distance 𝑟 from the axis travels along a circular path with linear (tangential) velocity 𝑣 = 𝑟𝜔. While all points share the same angular velocity and acceleration, their linear velocities differ depending on their distance from the axis. The direction of the linear velocity is always tangential to the circle at that point. This relationship simplifies analyzing rotational motion and is key to understanding the dynamics of rotating systems.
What is the relationship between angular velocity and linear velocity?
The linear velocity 𝑣 of a point on a rotating body is directly proportional to its distance 𝑟 from the axis of rotation and the angular velocity 𝜔, expressed as 𝑣 = 𝑟𝜔.
What is angular velocity (𝜔)?
Angular velocity is the rate at which an object rotates around a fixed axis, measured in radians per second (rad/s).
What is the direction of linear velocity in circular motion?
The linear velocity is always tangential to the circular path of the point.
What happens to linear velocity if the angular velocity is doubled?
If the angular velocity 𝜔 is doubled, the linear velocity 𝑣 also doubles, assuming the distance 𝑟 remains constant.
Why do all points on a rigid body have the same angular velocity?
All points on a rigid body share the same angular velocity because they rotate together around the same axis, maintaining their relative positions.
If the angular velocity of a body is doubled, what happens to its linear velocity?
a) It is halved.
b) It remains unchanged.
c) It is doubled.
d) It becomes four times larger.
Answer: c) It is doubled.
What is the direction of linear velocity in rotational motion?
a) Along the radius of the circle.
b) Tangential to the circular path.
c) Toward the center of the circle.
d) Opposite to the angular velocity.
Answer: b) Tangential to the circular path.
Which of the following quantities is the same for all points on a rigid body rotating about a fixed axis?
a) Linear velocity
b) Tangential acceleration
c) Angular velocity
d) Linear acceleration
Answer: c) Angular velocity
What is the linear velocity of a point located on the axis of rotation of a rotating body?
a) Zero
b) Equal to 𝑟𝜔
c) Infinite
d) Depends on the angular acceleration
Answer: a) Zero
Which of the following is not true for a point on a rotating rigid body?
a) Linear velocity is proportional to the radius.
b) Angular velocity is the same for all points.
c) Tangential acceleration is zero for uniform angular motion.
d) Linear velocity is inversely proportional to angular velocity.
Answer: d) Linear velocity is inversely proportional to angular velocity.