Angular Displacement, Speed, and Acceleration in Circular Motion

Angular Displacement, Speed, and Acceleration in Circular Motion

Uniform Circular Motion: Key Concepts

Uniform circular motion describes the movement of an object along a circular path at a constant speed. Several quantities are used to characterize this motion:

• Angular Position (θ): The angle, in radians, that an object has rotated from a reference direction.

• Arc Length (s): The distance traveled along the circular path.

The relationship between arc length and angular position is:

• $\theta = \frac{s}{r}$

• $s = r\theta$

For a complete revolution:

• $s = C = 2\pi r$ (where C is the circumference)

• $\theta = \frac{2\pi r}{r} = 2\pi$ radians$= 360^\circ$

• 1 radian $= \frac{180^\circ}{\pi} \approx 57.3^\circ$

Angular Speed and Frequency

Angular speed ($\omega$) measures how quickly an object rotates, defined as the rate of change of angular displacement:

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

• Units: radians per second (rad/s)

• Can also be expressed in terms of period (T) and frequency (f):

• $\omega = \frac{2\pi}{T} = 2\pi f$

Direction:

• $\omega$ is positive for counterclockwise (CCW) motion

• $\omega$ is negative for clockwise (CW) motion

Relationship Between Linear and Angular Speed

Linear speed ($v$) at a point on a rotating object is related to angular speed:

• $v = \omega r$

• Where $r$ is the distance from the axis of rotation

Example: For a wind turbine with $R = 50$ m and $f = 10$ RPM:

• $f = 10$ RPM $= 0.167$ rev/s

• $\omega = 2\pi f = 1.05$ rad/s

• $v = \omega R = 1.05 \times 50 = 52.5$ m/s

Earth's Rotation Example

Earth rotates once per day. The radius of Earth ($R_E$) is approximately $6.37 \times 10^6$ m.

• Frequency: $f = 1$ rev/day

• Period: $T = 1$ day $= 86400$ s

• Angular speed: $\omega = \frac{2\pi}{T}$

• Linear speed at the surface: $v = \omega R_E$

Angular Acceleration

Angular acceleration ($\alpha$) is the rate of change of angular velocity:

• $\alpha = \frac{\Delta \omega}{\Delta t}$

• Units: rad/s2

• For constant angular acceleration: $\Delta \omega = \alpha \Delta t$

Analogy to Linear Motion:

• Linear: $a = \frac{\Delta v}{\Delta t}$

• Circular: $\alpha = \frac{\Delta \omega}{\Delta t}$

• Linear: $x_f = x_i + v_i t + \frac{1}{2} a t^2$

• Circular: $\theta_f = \theta_i + \omega_i t + \frac{1}{2} \alpha t^2$

Example: Steel Ball on Roulette Wheel

• Radius: $R = 0.15$ m

• 2 revolutions in 1.25 s

• $\Delta \theta = 2 \times 2\pi = 4\pi$ rad

• $\omega = \frac{\Delta \theta}{\Delta t} = \frac{4\pi}{1.25} = 10.05$ rad/s

• RPM: $\frac{2}{1.25} \times 60 = 96$ RPM

Speed at Different Radii

For a shaft rotating at $f = 50$ rev/s ($\omega = 314$ rad/s):

• At $r = 0.1$ m: $v = \omega r = 314 \times 0.1 = 31.4$ m/s

Analogy Between Linear and Circular Motion

Key quantities in linear and circular motion:

Linear	Circular
x (position)	$\theta$ (angular position)
v (velocity)	$\omega$ (angular velocity)
a (acceleration)	$\alpha$ (angular acceleration)

Equations of motion:

• Linear: $x_f = x_i + v_i t + \frac{1}{2} a t^2$

• Circular: $\theta_f = \theta_i + \omega_i t + \frac{1}{2} \alpha t^2$

Acceleration in Rotational Motion

There are two types of acceleration for a rotating rigid body:

• Tangential acceleration ($a_t$): $a_t = \alpha r$ (due to change in speed)

• Centripetal acceleration ($a_c$): $a_c = \omega^2 r$ (due to change in direction)

• Total acceleration: $\vec{a}_{total} = \vec{a}_t + \vec{a}_c$

All points on a rotating object have the same $\omega$, but not the same $v$ (linear speed depends on $r$).

Summary Table: Linear vs. Circular Terms

Linear	Circular
s	$s = r\theta$
v	$v = r\omega$
a	$a_t = r\alpha$

Key Points

• Centripetal acceleration is always present in circular motion.

• Tangential acceleration may be zero or non-zero, depending on whether angular speed is changing.

• In circular motion, speed may be constant, but direction is always changing.

Additional info: These notes cover the essential concepts of rotational kinematics, including the relationships between angular and linear quantities, and the analogy to linear motion equations. They are suitable for college-level physics students studying Chapter 7: Rotational Motion.