10.1 Angular Acceleration - Teacher Physics 2e | 10.1 Angular Acceleration Open Stax (2023)

Uniform circular motion and gravityOnly uniform circular motion, that is, circular motion with constant velocity and thus constant angular velocity, is discussed. Think about that angular velocity$\mathrm{oh}$Defined as the time rate of change in angle$\mathrm{and}$:

Where$\mathrm{and}$is the rotation angle as followsFigure 10.3. The relationship between angular velocity$\mathrm{oh}$and line speed$v$is also defined inRotation Angle and Angular VelocityI

from

Where$\mathrm{right}$is the radius of curvature, and can also beFigure 10.3. By sign convention, counterclockwise is positive and clockwise is negative

Lectra10.3 The diagram shows uniform circular motion with some defined quantities.

Angular velocity is not constant when a skater pulls his arms back, when a child starts a carousel from rest, or when a computer's hard drive slows to a stop when it is turned off. In all these cases, there is aangular acceleration,in$\mathrm{oh}$Variety. The faster the change occurs, the greater the angular acceleration. angular acceleration$A$Defined as the rate of change of angular velocity. In equation form, the angular acceleration is expressed as follows:

Where$\mathrm{D}\mathrm{oh}$yeschange in angular velocityU$\mathrm{D}\mathrm{time}$is a change over time. The unit of angular acceleration is$\left(\text{rad/s}\right)\text{/S}$, from${\text{rad/s}}^2$. I$\mathrm{oh}$then increase$A$be positive. if$\mathrm{oh}$then reduce$A$is a negative number.

In the previous example, if the bike were standing on the wheels instead of standing upside down, it would first accelerate on the ground and then stop. The relationship between circular motion and linear motion needs to be explored. For example, it would be useful to know the relationship between linear acceleration and angular acceleration. Circular motion involves linear accelerationtouchMake a circle at the point of interest, as shownFigure 10.4. So there is a linear accelerationTangential acceleration $A_{\text{time}}$。

Lectra10.4 In circular motion, linear acceleration$A$, which occurs when the velocity magnitude changes:$A$touch sport. In circular motion, linear acceleration is also called tangential acceleration$A_{\text{time}}$。

Linear or tangential acceleration refers to a change in the magnitude of velocity, not its direction. we knowUniform circular motion and gravityThe centripetal acceleration of circular motion is$A_{\text{C}}$, referring to a change in the direction of the velocity, not a change in its magnitude. An object in circular motion will experience centripetal acceleration, as shown in the figureFigure 10.5. so,$A_{\text{time}}$U$A_{\text{C}}$perpendicular to each other and independent of each other. Tangential acceleration$A_{\text{time}}$directly related to angular acceleration$A$Relates to an increase or decrease in velocity, but not the direction of velocity.

Lectra10.5 centripetal acceleration$A_{\text{C}}$Occurs when the velocity direction is changed; it is perpendicular to the circular motion. Therefore, centripetal acceleration and tangential acceleration are perpendicular to each other.

Now we can find the exact relationship between the linear acceleration$A_{\text{time}}$and angular acceleration$A$. Since linear acceleration is proportional to a change in velocity magnitude, it is defined as (egone-dimensional kinematics) yes

Note that for circular motion$v=\mathrm{oh}$, so

radius$\mathrm{right}$is constant for circular motion, and so on$\text{D}（\mathrm{oh}）=\mathrm{right}（\mathrm{D}\mathrm{oh}）$. so,

By definition,$A=\frac{\mathrm{D}\mathrm{oh}}{\mathrm{D}\mathrm{time}}$. so,

from

These equations imply that linear acceleration is directly proportional to angular acceleration. The greater the angular acceleration, the greater the linear (tangential) acceleration and vice versa. The greater the angular acceleration of the driving wheels of the car, the greater the acceleration of the car. Radius is also important. The smaller the wheel, the smaller the linear acceleration for a given angular acceleration$A$。

example 10.2

Calculation of Motorcycle Wheel Angular Acceleration

This powerful motorcycle accelerates from 0 to 30.0 m/s (approximately 108 km/h) in 4.20 seconds. What is the angular acceleration of a wheel with a radius of 0.320 m? (lookFigure 10.6.)

Lectra10.6 The linear acceleration of the motorcycle follows the angular acceleration of the wheels.

strategy

We obtain information about the linear velocity of the motorcycle. In this way we can find its linear acceleration$A_{\text{time}}$. then the expression$A=\frac{A_{\text{time}}}{\mathrm{right}}$Can be used to find angular acceleration.

solution

The linear acceleration is

$$\begin{array}{lll}{A}_{\text{time}}& =& \frac{\mathrm{D}v}{\mathrm{D}\mathrm{time}}\\ & =& \frac{\text{30.0 m/s}}{\text{4,20 SEK}}\\ & =& \text{7.14}{\text{Mrs}}^2。\end{array}$$

10.15

We also know the wheel radius. input value$A_{\text{time}}$U$\mathrm{right}$Come in$A=\frac{A_{\text{time}}}{\mathrm{right}}$,we got

$$\begin{array}{lll}A& =& \frac{A_{\text{time}}}{\mathrm{right}}\\ & =& \frac{\text{7.14}{\text{Mrs}}^2}{\text{0,320 meters}}\\ & =& \text{22.3}{\text{rad/s}}^2。\end{array}$$

10.16

discuss

The radian unit is dimensionless and occurs in any relationship between angular and linear quantities.

So far we have defined three rotation quantities:$\mathrm{and},\mathrm{oh}$,you$A$. These quantities are similar to translational quantities$X,v$,you$A$。Table 10.1Shows rotation quantities, similar translation quantities, and the relationship between them.