Uniform circular motion (kinematics) Knowledge-Based Study

Study notes

1. What is Uniform Circular Motion?

   An object experiences uniform circular motion (UCM) when it travels along a circular path of fixed radius at a constant speed. Even though the speed remains unchanged, the direction of travel continuously changes at every instant.

   Because velocity is a vector quantity having both magnitude and direction, any change in direction means the object is accelerating, even when its speed is strictly constant.

   Period and Frequency

   • Period (T): The time required to complete one full revolution along the circular path. The standard SI unit is the second (s).
   • Frequency (f): The number of complete revolutions completed per unit time. The standard SI unit is the hertz (Hz), where 1 Hz = 1 s-1.

   Period and frequency are inversely related by the fundamental relationship:

   $T = \frac{1}{f}$

   Rotational Kinematic Quantities

   To describe position along a circular trajectory of radius r, it is convenient to use angular measures.

   Angular Displacement ($\Delta \theta$)

   When an object moves along an arc length $\Delta s$, the angle swept out relative to the center of the circle is its angular displacement:

   $\Delta \theta = \frac{\Delta s}{r}$

   Here, $\Delta \theta$ is measured in radians (rad). A complete revolution corresponds to $2\pi$ radians.

   Angular Velocity ($\omega$)

   The rate at which the angular position changes over time is the angular velocity:

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

   For a full revolution, where $\Delta \theta = 2\pi$ and $\Delta t = T$, the magnitude of the angular velocity is:

   $\omega = \frac{2\pi}{T} = 2\pi f$

   The standard SI unit for angular velocity is radians per second (rad/s).

   Connecting Linear and Angular Kinematics

   An object moving on a circular path has an instantaneous linear velocity that is always tangent to the circle. The magnitude of this vector is known as the tangential speed ($v$).

   Using the arc length relationship $\Delta s = r\Delta \theta$ and dividing by elapsed time $\Delta t$:

   $v = \frac{\Delta s}{\Delta t} = r\left(\frac{\Delta \theta}{\Delta t}\right) = r\omega$

   Alternatively, since the object travels a distance equal to the circumference $2\pi r$ during one period $T$:

   $v = \frac{2\pi r}{T} = 2\pi r f$

   Key Concept: For two points rotating on the same rigid platform, both share the exact same angular speed $\omega$, but a point farther from the rotation center has a greater tangential speed $v$ because of the larger radius.

   Direction and Magnitude of Acceleration

   In uniform circular motion, the acceleration vector does not change the speed of the object; instead, it solely changes the direction of the velocity vector. This acceleration is directed radially inward toward the center of the circular path and is called centripetal acceleration ($a_c$).

   Mathematical Expressions

   Using geometric vector subtraction or kinematic derivation, the magnitude of centripetal acceleration is expressed as:

   $a_c = \frac{v^2}{r}$

   Substituting the relation $v = \omega r$, we obtain alternative forms:

   $a_c = \omega^2 r = \frac{4\pi^2 r}{T^2}$

   Geometric Properties

   • The velocity vector is strictly tangential to the path.
   • The acceleration vector is strictly radial (pointing to the center).
   • The angle between the instantaneous velocity vector and the centripetal acceleration vector is always 90 degrees ($\frac{\pi}{2}\text{ rad}$).

   Summary of Kinematic Equations in UCM

   Uniform circular motion is a classic example of two-dimensional motion with constant speed but non-zero, continuously changing acceleration vector.

   • Speed: $v = \frac{2\pi r}{T} = \omega r$
   • Angular frequency / speed: $\omega = \frac{2\pi}{T} = 2\pi f$
   • Centripetal acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$

   Important Physical Takeaways

   Because the magnitude of velocity is constant, the tangential component of acceleration is zero ($a_t = 0$). The total linear acceleration is entirely centripetal: $a = a_c$. If the radius is doubled while keeping the tangential speed constant, the centripetal acceleration is halved. However, if the radius is doubled while maintaining constant angular speed, the centripetal acceleration doubles.