### learning target

At the end of this section, you can:

- Describe uniform circular motion.
- Explain non-uniform circular motion.
- Calculate the angular acceleration of an object.
- Observe the relationship between linear acceleration and angular acceleration.

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$\mathrm{and}$:

$$\mathrm{oh}=\frac{\mathrm{D}\mathrm{and}}{\mathrm{D}\mathrm{time}}\text{,}$$

10.1

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

$$v=\mathrm{oh}$$

10.2

from

$$\mathrm{oh}=\frac{v}{\mathrm{right}},$$

10.3

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:

$$A=\frac{\mathrm{D}\mathrm{oh}}{\mathrm{D}\mathrm{time}}\text{,}$$

10.4

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.

### example 10.1

#### Calculation of Bicycle Wheel Angular Acceleration and Deceleration

Suppose a teenager puts a bike on his back and spins the rear wheel from rest to a terminal angular velocity of 250 rpm in 5.00 seconds. (a) Calculate angular acceleration${\text{rad/s}}^{2}$. (b) Now if she hits the brakes, resulting in an angular acceleration of$\u2013\mathrm{87,3}\phantom{\rule{0ex}{0ex}}{\text{rad/s}}^{2}$How long does it take for the wheels to stop?

#### (a) Strategy

Angular acceleration can be found directly from its definition$A=\frac{\mathrm{D}\mathrm{oh}}{\mathrm{D}\mathrm{time}}$Because the final angular velocity and time are given. We saw$\mathrm{D}\mathrm{oh}$at 250 rpm,$\mathrm{D}\mathrm{time}$is 5.00 seconds.

#### Solution (1)

If we enter known information into the definition of angular acceleration, we will get

$$\begin{array}{lll}A& =& \frac{\mathrm{D}\mathrm{oh}}{\mathrm{D}\mathrm{time}}\\ & =& \frac{\text{250 t/min}}{\text{5.00 SEK}}\text{\u3002}\end{array}$$

10.5

because$\mathrm{D}\mathrm{oh}$Units are revolutions per minute (rpm), we want the default units to be${\text{rad/s}}^{2}$For angular acceleration, we need to convert$\mathrm{D}\mathrm{oh}$From revolutions per minute to rad/s:

$$\begin{array}{lll}\mathrm{D}\mathrm{oh}& =& \text{250}\frac{\text{Rotating speed}}{\text{minute}}\cdot \frac{\text{2\pi work}}{\text{Rotating speed}}\cdot \frac{\text{1 minute}}{\text{60 SEK}}\\ & =& \text{26.2}\frac{\text{rad}}{\text{S}}\u3002\end{array}$$

10.6

Enter this amount in the expression$A$,we got

$$\begin{array}{lll}A& =& \frac{\mathrm{D}\mathrm{oh}}{\mathrm{D}\mathrm{time}}\\ & =& \frac{\text{26.2 lines/sec}}{\text{5.00 SEK}}\\ & =& \text{5.24}{\text{rad/s}}^{2}\text{\u3002}\end{array}$$

10.7

#### (b) Strategy

In this section, we know angular acceleration and initial angular velocity. We can find the dwell time by solving the definition of angular acceleration using and$\mathrm{D}\mathrm{time}$, surrender

$$\mathrm{D}\mathrm{time}=\frac{\mathrm{D}\mathrm{oh}}{A}\text{\u3002}$$

10.8

#### Solution for (b)

As a result, the angular velocity decreases$\text{26.2 lines/sec}$(250 rpm) is zero, so$\mathrm{D}\mathrm{oh}$yes$\u2013\text{26.2 lines/sec}$,you$A$will give$\u2013\text{87,3}\phantom{\rule{0ex}{0ex}}{\text{rad/s}}^{2}$. so,

$$\begin{array}{lll}\mathrm{D}\mathrm{time}& =& \frac{\u2013\text{26.2 lines/sec}}{\u2013\text{87,3}\phantom{\rule{0ex}{0ex}}{\text{rad/s}}^{2}}\\ & =& \text{0,300 SEK.}\end{array}$$

10.9

#### discuss

Note that the angular acceleration as the girl turns the steering wheel is small and positive; it takes 5 seconds to develop a significant angular velocity. When he hits the brakes, the angular acceleration is high and negative. The angular velocity quickly goes to zero. In both cases, the relationship is similar to what happens in linear motion. For example, if you hit a brick wall, there is a large delay; large speed changes in short time intervals.

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 acceleration*touch*Make 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

$${A}_{\text{time}}=\frac{\mathrm{D}v}{\mathrm{D}\mathrm{time}}\text{\u3002}$$

10.10

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

$${A}_{\text{time}}=\frac{\mathrm{D}\left(\mathrm{oh}\right)}{\mathrm{D}\mathrm{time}}\text{\u3002}$$

10.11

radius$\mathrm{right}$is constant for circular motion, and so on$\text{D}\uff08\mathrm{oh}\uff09=\mathrm{right}\uff08\mathrm{D}\mathrm{oh}\uff09$. so,

$${A}_{\text{time}}=\mathrm{right}\frac{\mathrm{D}\mathrm{oh}}{\mathrm{D}\mathrm{time}}\text{\u3002}$$

10.12

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

$${A}_{\text{time}}=a,$$

10.13

from

$$A=\frac{{A}_{\text{time}}}{\mathrm{right}}\u3002$$

10.14

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

$$\begin{array}{lll}{A}_{\text{time}}& =& \frac{\mathrm{D}v}{\mathrm{D}\mathrm{time}}\\ & =& \frac{\text{30.0 m/s}}{\text{4,20 SEK}}\\ & =& \text{7.14}\phantom{\rule{0ex}{0ex}}{\text{Mrs}}^{2}\u3002\end{array}$$

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

$$\begin{array}{lll}A& =& \frac{{A}_{\text{time}}}{\mathrm{right}}\\ & =& \frac{\text{7.14}\phantom{\rule{0ex}{0ex}}{\text{Mrs}}^{2}}{\text{0,320 meters}}\\ & =& \text{22.3}\phantom{\rule{0ex}{0ex}}{\text{rad/s}}^{2}\u3002\end{array}$$

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.

turn around | translate eel | relation |
---|---|---|

$$\mathrm{and}$$ | $$X$$ | $$\mathrm{and}=\frac{X}{\mathrm{right}}$$ |

$$\mathrm{oh}$$ | $$v$$ | $$\mathrm{oh}=\frac{v}{\mathrm{right}}$$ |

$$A$$ | $$A$$ | $$A=\frac{{A}_{\mathrm{time}}}{\mathrm{right}}$$ |

table 10.1 rotation and translation

### Making Connections: An Experiment to Take Home

Sit in a swivel chair with your feet on the floor. Lift one leg so it is not bent (extended). Using the other leg, push yourself onto the floor and start turning. Stop pushing the floor with your legs and let the chair spin. From where you started, plot the leg's angle, angular velocity, and angular acceleration over time as three separate graphs. Estimate the size of these quantities.

### check your understanding

Angular acceleration is a vector that has both magnitude and direction. How do we indicate its size and orientation? for example.

#### solution

The magnitude of the angular acceleration is$A$The most common is the unit${\text{rad/s}}^{2}$. The direction of angular acceleration along a fixed axis is indicated by a + or - sign, just as the direction of linear acceleration in one dimension is indicated by a + or - sign. Take the example of a gymnast doing a front flip. Her angular momentum will be parallel to the mat and to her left. The magnitude of its angular acceleration is proportional to its angular velocity (rotational speed) and its moment of inertia around the axis of rotation.

### ladybug revolution

Explore rotational motion with Ladybug. Turn the dial to change the angle, or choose a constant angular velocity or angular acceleration. Use vectors or graphs to understand how circular motion relates to the bug's x,y position, velocity, and acceleration.

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