Rotational Motion and Gravity Notes

Chapter 7: Rotational Motion and The Law of Gravity

Radian Measure

• The radian is a unit of angular measure.

• It's defined as the arc length $s$ along a circle divided by the radius $r$.

Radians in Context

• Comparison of degrees and radians:

  • $360^{\circ} = 2\pi$ radians

  • $1 \text{ rad} = 57.3^{\circ}$

• Converting from degrees to radians:

  • $\theta \text{ [rad]} = \frac{\pi}{180^{\circ}} \theta \text{ [degrees]}$

Angular Displacement

• The axis of rotation is at the center of the disk.

• A fixed reference line is needed to measure angular displacement.

• During time $t$, the reference line moves through an angle $\_theta$.

• Angular position: $\_theta$ (measured in a counterclockwise direction).

Rigid Body

• Every point on a rigid object undergoes circular motion about the point $O$.

• All parts of the object rotate through the same angle during the same time.

• A rigid body implies that each part of the body is fixed in position relative to all other parts of the body.

Angular Displacement (Continued)

• Angular displacement is defined as the change in angular position.

• The unit of angular displacement is the radian.

• Each point on the object undergoes the same angular displacement.

Average Angular Speed

• The average angular speed $\omega$ of a rotating rigid object is the ratio of the angular displacement to the time interval:

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

Angular Speed

• The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero.

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

• Units of angular speed are radians/sec (rad/s).

• Speed is positive if $\_theta$ is increasing (counterclockwise).

• Speed is negative if $\_theta$ is decreasing (clockwise).

Average Angular Acceleration

• The average angular acceleration $\alpha$ of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

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

Angular Acceleration

• Units of angular acceleration are rad/s².

• Positive angular accelerations are in the counterclockwise direction, and negative accelerations are in the clockwise direction.

• When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration.

Angular Acceleration (Final)

• The sign of the acceleration does not have to be the same as the sign of the angular speed.

• The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero.

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

Analogies Between Linear and Rotational Motion

• Linear Motion (Constant Acceleration):

  • Velocity: $v = v_0 + at$

  • Displacement: $\Delta x = v_0t + \frac{1}{2}at^2$

  • Velocity-Displacement: $v^2 = v_0^2 + 2a\Delta x$

• Rotational Motion (Constant Angular Acceleration):

  • Angular Velocity: $\omega = \omega_0 + \alpha t$ [7.7]

  • Angular Displacement: $\Delta \theta = \omega_0 t + \frac{1}{2}\alpha t^2$ [7.8]

  • Angular Velocity-Displacement: $\omega^2 = \omega_0^2 + 2\alpha \Delta \theta$ [7.9]

Example Problem: Airplane Propeller

• An airplane propeller slows down from an initial angular speed of 12.5 revolutions/sec to a final angular speed of 5.00 revolutions/sec.

• During this process, the propeller rotates through 21.0 revolutions.

• If the angular acceleration is constant, what is the angular acceleration in radians per second squared?

• Solution:

  • Convert the angular displacement to radians and the angular speeds to radians/sec:

    • $\theta = (21.0 \text{ rev}) \cdot (\frac{2\pi \text{ rad}}{1 \text{ rev}}) = 42.0\pi \text{ rad}$

    • $\omega_i = (12.5 \text{ rev/s}) \cdot (\frac{2\pi \text{ rad}}{1 \text{ rev}}) = 25.0\pi \text{ rad/s}$

    • $\omega_f = (5.00 \text{ rev/s}) \cdot (\frac{2\pi \text{ rad}}{1 \text{ rev}}) = 10.0\pi \text{ rad/s}$

  • Use the equation: $\omega<em>f^2 = \omega</em>i^2 + 2\alpha \Delta \theta$

    • $(10.0\pi \text{ rad/s})^2 = (25.0\pi \text{ rad/s})^2 + 2 \cdot \alpha \cdot (42.0\pi \text{ rad})$

    • $\alpha = -6.25 \pi \text{ rad/s}^2$

Relationship Between Angular and Linear Quantities

• Displacements

• Speeds

• Accelerations

• Every point on a rotating object has the same angular motion.

• Every point on a rotating object does not have the same linear motion.

Relations Between Angular and Linear Quantities

1. Arc Length: $\Delta s = r \Delta \theta$

2. Dividing both sides by $\Delta t$:

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

   • Taking the limit as $\Delta t$ approaches 0:

     • $\lim<em>{\Delta t \to 0} \frac{\Delta s}{\Delta t} = r \lim</em>{\Delta t \to 0} \frac{\Delta \theta}{\Delta t}$

     • $v_t = r\omega$

     • The tangential speed of a point on a rotating object equals the distance of that point from the axis of rotation multiplied by the angular speed.

3. Divide both sides by $\Delta t$:

   • $\frac{\Delta v_t}{\Delta t} = r \frac{\Delta \omega}{\Delta t}$

   • Taking the limit as $\Delta t \to 0$:

     • $\lim<em>{\Delta t \to 0} \frac{\Delta v</em>t}{\Delta t} = r \lim_{\Delta t \to 0} \frac{\Delta \omega}{\Delta t}$

     • $a_t = r\alpha$

     • The tangential acceleration of a point on a rotating object equals the distance of that point from the axis of rotation multiplied by the angular acceleration.

Centripetal Acceleration

• An object traveling in a circle, even at constant speed, has an acceleration.

• Centripetal acceleration is due to the change in the direction of the velocity.

Centripetal Acceleration (Circular Motion at Constant Speed)

• Centripetal means "center-seeking."

• The direction of the velocity changes.

• The acceleration is directed toward the center of the circle of motion.

Derivation of Centripetal Acceleration

• Assume the car moves at a constant speed, so $v<em>1$ and $v</em>2$ differ only in direction.

• $a<em>{av} = \frac{\Delta v}{\Delta t} = \frac{v</em>2 - v<em>1}{t</em>2 - t_1}$

• When $\Delta t$ is very small, $\Delta s$ and $\Delta \theta$ are also very small. $v<em>1$ is almost parallel to $v</em>2$, and $\Delta v$ is approximately perpendicular to them, pointing toward the center of the circle.

• In the limit as $\Delta t \to 0$, $\Delta v$ points exactly toward the center of the circle, and $a_{av} \to a$ (instantaneous acceleration).

• Since $\Delta v = v \frac{\Delta s}{r}$. Dividing both sides by $\Delta t$:

  • $\frac{\Delta v}{\Delta t} = \frac{v}{r} \frac{\Delta s}{\Delta t}$

  • Taking the limit as $\Delta t \to 0$:

    • $\lim<em>{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{v}{r} \lim</em>{\Delta t \to 0} \frac{\Delta s}{\Delta t}$

    • $a<em>c = \frac{v^2}{r}$, where $a</em>c$ is called centripetal acceleration.

• Therefore, the magnitude of $a_c$ is:

  • $a_c = \frac{v^2}{r} = r\omega^2$

Tangential Acceleration

• The previous derivations concern circular motion at constant speed.

• When an object moves in a circle but is speeding up or slowing down, a tangential acceleration $a_t$ is also present.

• Thus, for an object moving in a circle, the total acceleration is:

  • $a<em>{\text{total}} = a</em>t + a_c$

Centripetal Acceleration (Final)

• The magnitude of the centripetal acceleration is given by $a_c = \frac{v^2}{r}$

• This direction is toward the center of the circle.

Centripetal Acceleration and Angular Velocity

• The angular velocity and the linear velocity are related $(v = \omega r)$.

• The centripetal acceleration can also be related to the angular velocity: $a_c = r\omega^2$

Total Acceleration

• The tangential component of the acceleration is due to changing speed.

• The centripetal component of the acceleration is due to changing direction.

• Total acceleration can be found from these components.

Example Problem: Race Car

• A race car accelerates uniformly from a speed of 40.0 m/s to a speed of 60.0 m/s in 5.00 s while traveling around a circular track of radius $4.00 \times 10^2$ m.

• When the car reaches a speed of 50.0 m/s, find:

  • (a) the magnitude of the car's centripetal acceleration;

  • (b) the angular speed;

  • (c) the magnitude of the tangential acceleration; and

  • (d) the magnitude of the total acceleration.

Solution: Race Car Problem

• (a) Centripetal acceleration:

  • $a_c = \frac{v^2}{r} = \frac{(50.0 \text{ m/s})^2}{4.00 \times 10^2 \text{ m}} = 6.25 \text{ m/s}^2$

• (b) Angular speed:

  • $\omega = \frac{v}{r} = \frac{50.0 \text{ m/s}}{4.00 \times 10^2 \text{ m}} = 0.125 \text{ rad/s}$

• (c) Tangential acceleration (uniform acceleration):

  • $a_t = \frac{\Delta v}{\Delta t} = \frac{60.0 \text{ m/s} - 40.0 \text{ m/s}}{5.00 \text{ s}} = 4.00 \text{ m/s}^2$

• (d) Total acceleration:

  • Since $a<em>t$ is tangential to the track and $a</em>c$ is pointing toward the center of the track:

    • $a<em>{\text{total}} = \sqrt{a</em>t^2 + a_c^2} = \sqrt{(4.00 \text{ m/s}^2)^2 + (6.25 \text{ m/s}^2)^2} = 7.42 \text{ m/s}^2$

This total acceleration indicates the net effect of both the tangential and centripetal accelerations acting on the object as it moves along a curved path. Understanding the components of total acceleration is crucial for analyzing motion in a circular path, as they demonstrate how forces interact to influence the object's trajectory.

Centripetal Force

• General equation: $F<em>c = ma</em>c = m\frac{v^2}{r}$

• If the force vanishes, the object will move in a straight line tangent to the circle of motion.