6.1 Rotation Angle and Angular Velocity

Rotation Angle and Arc Length

Arc length in circular motion

When an object moves along a circular path, the distance it covers along the curve is called the arc length ($s$). Arc length is measured in standard length units like meters.

Arc length depends on two things: how big the circle is (the radius $r$) and how far the object has rotated (the angle $\theta$). The relationship is:

$s = r\theta$

where $\theta$ must be in radians for this formula to work. This is a detail that trips people up on exams: if you're given degrees, convert to radians first.

• Example: A point on a circle with radius 2 m rotates through $\frac{\pi}{3}$ radians. The arc length is $s = 2 \cdot \frac{\pi}{3} \approx 2.09$ m.

Rotation angle and radius of curvature

The rotation angle ($\theta$) describes how far an object has turned along its circular path. It can be measured in radians or degrees, with one full rotation equal to $2\pi$ radians or 360°.

• A quarter turn = $\frac{\pi}{2}$ rad = 90°
• A half turn = $\pi$ rad = 180°
• A full turn = $2\pi$ rad = 360°

To convert between degrees and radians: $\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$

The radius of curvature ($r$) is the distance from the center of the circle to any point on the path. For the same rotation angle, a larger radius produces a longer arc length. Picture two runners on a track: the runner in the outer lane covers more ground per lap than the runner in the inner lane, even though both complete the same angular rotation.

Angular Velocity and Its Relationship to Linear Velocity

Angular velocity

Angular velocity ($\omega$) measures how fast an object rotates. It's defined as the change in rotation angle divided by the elapsed time:

$\omega = \frac{\Delta\theta}{\Delta t}$

The standard unit is radians per second (rad/s).

• Example: A wheel completes a half turn ($\pi$ radians) in 2 seconds. Its angular velocity is $\omega = \frac{\pi}{2} \approx 1.57$ rad/s.

Linear velocity and its connection to angular velocity

Linear velocity ($v$) describes how fast a point on the circle is actually moving through space, measured in m/s. It's defined as:

$v = \frac{\Delta s}{\Delta t}$

Since $s = r\theta$, you can connect the two velocities directly:

$v = r\omega$

This equation is one of the most useful in this unit. It tells you that linear velocity depends on both how fast the object spins and how far it is from the center.

• Two points on a spinning CD have the same angular velocity $\omega$, but a point on the outer edge has a larger $r$, so it moves faster through space than a point near the center.
• A merry-go-round spinning at 0.5 rad/s: a rider sitting 3 m from the center has $v = 3 \times 0.5 = 1.5$ m/s, while a rider at 1 m from the center has $v = 0.5$ m/s.

Applications of rotational motion

Wheels and gears: The linear speed of a point on a wheel's rim is $v = r\omega$. This is why larger wheels cover more ground per rotation. In gear systems, meshing gears of different sizes lets you trade angular velocity for torque (or vice versa). A small gear driving a large gear reduces the rotation speed but increases torque.

Celestial bodies: Planets and moons rotate about their own axes with measurable angular velocities. Earth completes one full rotation ($2\pi$ rad) in about 24 hours, giving it an angular velocity of roughly $7.27 \times 10^{-5}$ rad/s. Jupiter spins much faster, completing a rotation in about 10 hours. The apparent motion of the Sun across the sky is a direct result of Earth's rotation.

Rotational Dynamics and Energy (Preview)

These topics go beyond rotation angle and angular velocity, but they'll come up later in the course. For now, just know the basic vocabulary:

• Torque is the rotational equivalent of force. It causes changes in angular velocity.
• Moment of inertia is the rotational equivalent of mass. It describes how much an object resists changes in its rotation.
• Rotational kinetic energy depends on moment of inertia and angular velocity.
• Angular momentum is conserved when no external torques act on a system, just as linear momentum is conserved without external forces.