Rotation: Rotational Variables and Their Relationship to Linear Motion

Chapter 5: Rotation

Outline

• Rotational variables

• Relating the linear and angular variables

• Kinetic energy of rotation

• Calculating the rotational inertia

• Torque

• Newton's second law for rotation

• Work and rotational kinetic energy

Rotational Variables

Translation vs. Rotation

• Translation: An object moves along a straight or curved line.

• Rotation: An object turns about an axis.

A rigid body is a body that can rotate with all its parts locked together and without any change in its shape. A fixed axis (rotation axis) means that the rotation occurs about an axis that does not move.

• In pure rotation (angular motion):

  • Every point of the body moves in a circle whose center lies on the axis of rotation.

  • Every point moves through the same angle during a particular time interval.

Angular Position (θ)

The angular position of the reference line is the angle of the line relative to a fixed direction, taken as the zero angular position (the positive direction of the x-axis).

From geometry, the angular position θ is given by:

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

• s: Length of a circular arc that extends from the x-axis to the reference line

• r: Radius of the circle

1 revolution (from the zero angular position) = $2\pi$ rad. If the reference line completes two revolutions: $\theta = 4\pi$ rad.

Angular Displacement (Δθ)

If the body rotates about the rotation axis, changing the angular position of the reference line from $\theta_1$ to $\theta_2$, the body undergoes an angular displacement:

$\Delta \theta = \theta_2 - \theta_1$

• $\Delta \theta > 0$ in counterclockwise direction

• $\Delta \theta < 0$ in clockwise direction

Angular Velocity (ω)

• The average angular velocity of the body during $\Delta t$ is:

  $\omega_{avg} = \frac{\Delta \theta}{\Delta t}$

• The instantaneous angular velocity is:

  $\omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$

Angular Acceleration (α)

• If the angular velocity $\omega$ is not constant, the body has an angular acceleration:

• The average angular acceleration is:

  $\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$

• The instantaneous angular acceleration is:

  $\alpha = \lim_{\Delta t \to 0} \frac{\Delta \omega}{\Delta t} = \frac{d\omega}{dt}$

Example Problem 1: Rotational Motion of a Disk

• A disk rotates about its central axis. The angular position $\theta$ as a function of time is given by $\theta = -1.00t - 0.200t^2 + 0.0100t^3$ (with $t$ in seconds, $\theta$ in radians).

• Tasks include graphing $\theta$ vs. $t$, finding the minimum value, and describing the disk's motion.

Key Steps:

• To find the angular position at a given time, substitute $t$ into the equation for $\theta$.

• To find the minimum value, set the derivative $d\theta/dt$ to zero and solve for $t$.

• To find angular velocity, use $\omega = d\theta/dt$.

Description: The disk starts with a positive angular position, turns clockwise while slowing, stops, and then turns counterclockwise, with its angular position eventually becoming positive again.

Example Problem 2: Angular Acceleration of a Spinning Top

• A child's top is spun with angular acceleration $\alpha = 5t^2 - 4t$ (with $t$ in seconds, $\alpha$ in rad/s$^2$). At $t=0$, the top has angular velocity 5 rad/s and angular position 2 rad.

• (a) To find angular velocity $\omega(t)$, integrate $\alpha$ with respect to $t$:

$\omega(t) = \int \alpha \, dt = \int (5t^2 - 4t) \, dt = \frac{5}{3}t^3 - 2t^2 + C$

Using the initial condition $\omega(0) = 5$ rad/s, $C = 5$.

$\omega(t) = \frac{5}{3}t^3 - 2t^2 + 5$

• (b) To find angular position $\theta(t)$, integrate $\omega(t)$ with respect to $t$:

$\theta(t) = \int \omega(t) \, dt = \int \left(\frac{5}{3}t^3 - 2t^2 + 5\right) dt = \frac{5}{12}t^4 - \frac{2}{3}t^3 + 5t + C'$

Using $\theta(0) = 2$ rad, $C' = 2$.

$\theta(t) = \frac{5}{12}t^4 - \frac{2}{3}t^3 + 5t + 2$

Relating the Linear and Angular Variables

It is important to relate the linear variables ($s$, $v$, $a$) to the angular variables ($\theta$, $\omega$, $\alpha$).

The Position $s$

• A point in a rigid rotating body, at a perpendicular distance $r$ from the rotation axis, moves in a circle with radius $r$.

• If the body rotates through an angle $\theta$, the point moves along an arc with length $s$ given by:

$s = \theta r$

The Speed $v$

• The linear velocity $\vec{v}$ of the point is tangent to the circle; the linear speed $v$ is given by:

$v = \omega r$

• If $\omega$ is constant, $v$ is constant and the point moves in uniform circular motion.

• The period $T$ of the motion for the point and the body is:

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

The Acceleration $a$

• The tangential component of acceleration (responsible for changes in the magnitude of $v$):

$a_t = \alpha r$

• In addition, a particle moving in a circular path has a radial (centripetal) component of linear acceleration:

$a_r = \frac{v^2}{r} = \omega^2 r$

Example Problem 3: Rotational Motion of a Large Ring

• A large horizontal ring rotates about a vertical axis with a radius of 31 m. The angular position $\theta$ is given by $\theta = 6.30t - 0.600t^2$ (rad).

• Tasks include finding angular speed, linear speed, tangential acceleration, and radial acceleration at a specific time.

Key Steps:

• Angular speed $\omega = d\theta/dt$

• Linear speed $v = \omega r$

• Tangential acceleration $a_t = \alpha r$, where $\alpha = d\omega/dt$

• Radial acceleration $a_r = \omega^2 r$

• Total acceleration is the vector sum of tangential and radial components:

$a = \sqrt{a_t^2 + a_r^2}$

Summary Table: Linear and Angular Variables

Linear Variable	Angular Variable	Relationship
Displacement ($s$)	Angular displacement ($\theta$)	$s = r\theta$
Velocity ($v$)	Angular velocity ($\omega$)	$v = r\omega$
Acceleration ($a$)	Angular acceleration ($\alpha$)	$a_t = r\alpha$ (tangential), $a_r = r\omega^2$ (radial)

Example Applications:

• Calculating the speed of a point on a rotating disk.

• Determining the acceleration of a rider on a rotating amusement park ride.

Additional info: The notes above cover the first two main topics in the chapter outline: rotational variables and the relationship between linear and angular variables. The remaining topics (kinetic energy of rotation, rotational inertia, torque, Newton's second law for rotation, and work/rotational kinetic energy) would follow in subsequent sections of the chapter.