5.1 Rotational Kinematics

Rotational kinematics describes the motion of objects rotating about an axis. It uses angular measurements like displacement, velocity, and acceleration to analyze circular motion, similar to linear kinematics for straight-line motion.

The study of rotating objects requires understanding both the rigid nature of these systems and how different points on the same object move differently during rotation. We analyze this motion using angular quantities that parallel their linear counterparts.

Angular motion measurements

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Angular displacement in radians

Angular displacement measures how far an object has rotated around an axis, measured in radians. A radian is the angle subtended when the arc length equals the radius of the circle. Angular displacement is the change in angular position and is calculated by $\Delta \theta = \theta - \theta_0$, where $\theta_0$ is the initial angular position and $\theta$ is the final angular position.

• Rigid systems hold their shape but different points move in different directions during rotation, meaning we cannot model them as single particles
• The direction of angular displacement (clockwise or counterclockwise) is assigned a positive or negative value to track rotation direction
• Systems can be treated as single objects if the rotation about an axis is well described by the motion of its center of mass
• When considering Earth's revolution around the Sun, Earth's rotation about its axis becomes negligible

Average angular velocity

Average angular velocity represents how quickly an object's angular position changes over time. This concept parallels linear velocity but applies to rotational motion.

• Calculated using the equation: $\omega_{avg} = \frac{\Delta \theta}{\Delta t}$
• The unit for angular velocity is radians per second (rad/s)
• Angular velocity indicates both the rate of rotation and its direction
• A constant angular velocity means an object rotates through equal angles in equal time intervals

Average angular acceleration

Average angular acceleration describes how an object's rotation speed changes over time. When angular velocity increases or decreases, angular acceleration is present.

• Determined by the equation: $\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$
• The unit for angular acceleration is radians per second squared (rad/s²)
• Positive angular acceleration increases the angular velocity in the positive direction
• Negative angular acceleration decreases the angular velocity or increases it in the negative direction

Angular vs linear motion

Angular motion equations closely parallel linear motion equations. The mathematical relationships between displacement, velocity, and acceleration work similarly in both domains, just with different units.

• Angular displacement, velocity, and acceleration around one axis are analogous to their linear counterparts in one dimension
• Key mathematical relationships for constant angular acceleration include:
  • $\omega = \omega_0 + \alpha t$
  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$
• We can analyze graphs of angular displacement, angular velocity, and angular acceleration versus time. The slope of a $\theta$ vs. $t$ graph gives angular velocity, the slope of a $\omega$ vs. $t$ graph gives angular acceleration, the area under a $\omega$ vs. $t$ graph gives angular displacement, and the area under an $\alpha$ vs. $t$ graph gives change in angular velocity.

Practice Problem 1: Angular Displacement

Solution:

1. We need to find the angular displacement after 6 seconds.
2. We can use the equation: $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
3. Given:

• Initial angular displacement $\theta_0 = 0$ (starting from rest)
• Initial angular velocity $\omega_0 = 0$ (starting from rest)
• Angular acceleration $\alpha = 2.5$ rad/s²
• Time $t = 6$ s

1. Substituting: $\theta = 0 + 0 + \frac{1}{2} \times 2.5 \times 6^2$ $\theta = \frac{1}{2} \times 2.5 \times 36$ $\theta = 45$ radians
2. To convert to revolutions, we divide by $2\pi$: Number of revolutions = $\frac{45}{2\pi} \approx 7.16$ revolutions

The wheel completes approximately 7.16 revolutions in the first 6 seconds.

Practice Problem 2: Angular Velocity

Solution:

1. We can use the equation: $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

2. Given:

• Initial angular velocity $\omega_0 = 25$ rad/s
• Final angular velocity $\omega = 0$ rad/s (stopped)
• Angular displacement $\theta - \theta_0 = 125$ rad

1. Substituting: $0^2 = 25^2 + 2\alpha \times 125$ $0 = 625 + 250\alpha$ $-625 = 250\alpha$ $\alpha = -2.5$ rad/s²
2. The negative sign indicates that the acceleration is in the opposite direction of the initial velocity, which makes sense since the flywheel is slowing down.

The angular acceleration of the flywheel is -2.5 rad/s².

Practice Problem 3: Using Angular Velocity and Acceleration

Solution:

1. We can use the equation: $\omega = \omega_0 + \alpha t$
2. Given:

• Initial angular velocity $\omega_0 = 4$ rad/s
• Angular acceleration $\alpha = 3$ rad/s²
• Time $t = 5$ s

1. Substituting: $\omega = 4 + (3)(5) = 19$ rad/s

The final angular velocity of the disk is 19 rad/s.