10.2 Kinematics of Rotational Motion

Rotational motion kinematics describes how objects move in circular paths. It's like linear motion, but with a twist! We use angular displacement, velocity, and acceleration to track spinning objects, just as we use position, speed, and acceleration for straight-line motion.

Comparing linear and rotational motion helps us understand both better. While a car's straight-line movement is linear, a Ferris wheel's circular motion is rotational. Both types of motion have similar equations, but use different variables and units to describe movement.

Rotational Motion Kinematics

Key variables in rotational kinematics

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• Angular displacement ([object Object],[object Object]) represents the change in angular position of a rotating object measured in radians (rad) or degrees (°)
  • Analogous to linear displacement ($\Delta x$) in linear motion
  • Example: a door opening from 0° to 90° has an angular displacement of 90° or $\frac{\pi}{2}$ rad
• Angular velocity ($\omega$) represents the rate of change of angular displacement over time measured in radians per second (rad/s) or degrees per second (°/s)
  • Analogous to linear velocity ($v$) in linear motion
  • Example: a ceiling fan rotating at a constant rate of 120 rpm (revolutions per minute) has an angular velocity of $4\pi$ rad/s or 720 °/s
• Angular acceleration ([object Object],[object Object]) represents the rate of change of angular velocity over time measured in radians per second squared (rad/s²) or degrees per second squared (°/s²)
  • Analogous to linear acceleration ($a$) in linear motion
  • Example: a washing machine that increases its spin speed from 0 to 1200 rpm in 30 seconds has an angular acceleration of $4\pi$ rad/s² or 720 °/s²
• Radius of rotation ([object Object],[object Object]) is the distance from the axis of rotation to the point of interest on the rotating object
  • Determines the relationship between linear and angular quantities
  • Example: a point on the rim of a bicycle wheel has a larger radius of rotation than a point near the hub
• Tangential velocity ([object Object],[object Object]) is the linear velocity of a point on a rotating object related to angular velocity by $v_t = r\omega$
  • Directed tangent to the circular path of the point
  • Example: a point on the equator of the Earth has a tangential velocity of about 1670 km/h due to the Earth's rotation
• Tangential acceleration ([object Object],[object Object]) is the linear acceleration of a point on a rotating object related to angular acceleration by $a_t = r\alpha$
  • Directed tangent to the circular path of the point
  • Example: a point on a spinning CD experiences tangential acceleration as the CD player changes its rotation speed
• Centripetal acceleration is the acceleration of a point on a rotating object directed towards the center of rotation, given by $a_c = r\omega^2$

Problem-solving with rotational equations

• $\Delta\theta = \omega_0 t + \frac{1}{2}\alpha t^2$ calculates angular displacement when initial angular velocity ($\omega_0$), angular acceleration, and time ($t$) are known
  • Similar to the linear motion equation $\Delta x = v_0 t + \frac{1}{2}a t^2$
  • Example: a flywheel with an initial angular velocity of 10 rad/s and an angular acceleration of 2 rad/s² will have an angular displacement of 80 rad after 6 seconds
• $\omega_f = \omega_0 + \alpha t$ calculates final angular velocity ($\omega_f$) when initial angular velocity, angular acceleration, and time are known
  • Similar to the linear motion equation $v_f = v_0 + a t$
  • Example: a potter's wheel with an initial angular velocity of 50 rpm and an angular acceleration of 10 rpm/s will have a final angular velocity of 110 rpm after 6 seconds
• $\omega_f^2 = \omega_0^2 + 2\alpha\Delta\theta$ calculates final angular velocity when initial angular velocity, angular acceleration, and angular displacement are known
  • Similar to the linear motion equation $v_f^2 = v_0^2 + 2a\Delta x$
  • Example: a wind turbine with an initial angular velocity of 2 rad/s, an angular acceleration of 0.5 rad/s², and an angular displacement of 10 rad will have a final angular velocity of about 3.32 rad/s
• $\Delta\theta = \frac{1}{2}(\omega_0 + \omega_f)t$ calculates angular displacement when initial angular velocity, final angular velocity, and time are known
  • Similar to the linear motion equation $\Delta x = \frac{1}{2}(v_0 + v_f)t$
  • Example: a rotating platform that starts at 15 rpm and ends at 45 rpm after 10 seconds will have an angular displacement of 300 revolutions or 1200$\pi$ rad

Linear vs rotational motion comparisons

• Similarities between linear and rotational motion:
  1. Both describe the motion of objects and have displacement, velocity, and acceleration
  2. Equations for linear and rotational motion have similar forms (e.g., $\Delta x = v_0 t + \frac{1}{2}a t^2$ and $\Delta\theta = \omega_0 t + \frac{1}{2}\alpha t^2$)
• Differences between linear and rotational motion:
  1. Linear motion describes motion along a straight line (e.g., a train moving on a track), while rotational motion describes motion around an axis (e.g., a Ferris wheel)
  2. Linear motion uses variables such as position ($x$), velocity ($v$), and acceleration ($a$), while rotational motion uses angular displacement ($\theta$), angular velocity ($\omega$), and angular acceleration ($\alpha$)
  3. Units for linear motion are typically meters (m) and seconds (s), while units for rotational motion are radians (rad) or degrees (°) and seconds (s)
• Examples of linear motion:
  1. A bullet fired from a gun
  2. An elevator moving between floors
• Examples of rotational motion:
  1. The blades of a helicopter
  2. A figure skater spinning on the ice

Rotational dynamics

• Moment of inertia is a measure of an object's resistance to rotational acceleration, analogous to mass in linear motion
• Torque is the rotational equivalent of force, causing angular acceleration in rotating objects
• Angular momentum is the rotational analog of linear momentum, describing the tendency of a rotating object to maintain its rotation