# 6.3 Rotation movement - Physics | OpenStax (2023)

### some learning objectives

By the end of this section, you will be able to:

• Describe the rotational kinematics variables and equations and relate them to the corresponding linear variables
• Describe torque and handling
• Solve problems related to torque and rotational kinematics.

### teacher support

#### teacher support

• (4) Scientific concepts. Students understand and apply the laws of motion in a variety of situations. Students must:
• (C)Analyze and describe accelerated motion in 2D using equations, including projectile and circular examples.
• (Fuego)Calculate the effects of forces on objects, including the laws of inertia, the relationship between force and acceleration, and the properties of force pairs between objects.

Additionally, the High School Physics Lab Manual addresses this section titled "Circular and Rotational Motion" in the lab along with the following criteria:

• (4) Scientific concepts. Students understand and apply the laws of motion in a variety of situations. Students must:
• (Fuego) Calculate the effects of forces on objects, including the laws of inertia, the relationship between force and acceleration, and the properties of force pairs between objects.

### some keywords

 angular acceleration Rotation movement kinematics lever tangential acceleration torque

### teacher support

#### teacher support

[knowledge][OL]Review linear equations of motion.

### misinterpreted warning

Students may be confused by deceleration and acceleration increasing in negative directions.

In the section on uniform circular motion, we discussed uniform circular motion, and therefore uniform circular motion. But sometimes the angular velocity is not constant: the rotational motion can speed up, slow down, or reverse direction. Angular velocity is not constant when a spinning skater pulls on your arm, when a child pushes on a merry-go-round to make it spin, or when a CD is turned off and brakes to a stop. In all these cases,angular accelerationoccurs due to angular velocity$OhOh$Variety. The faster the change occurs, the greater the angular acceleration. angular acceleration$ININ$is the rate of change of angular velocity. In the equation, the average angular acceleration is

$IN= commandOh commandton , IN= commandOh commandton ,$

Where$commandOh commandOh$is the change in angular velocity and$commandton commandton$is the change of time. The unit of angular acceleration is (rad/s)/s or rad/s2.and$OhOh$so get up$ININ$Be positive. Yeah$OhOh$so it falls$ININ$is negative Remember that counterclockwise is positive and clockwise is negative. For example, skaters inFigure 6.9Viewed from above, it rotates counterclockwise, so its angular velocity is positive. Acceleration will be negative, for example, when an object rotating counterclockwise slows down. It will be positive when an object rotating counterclockwise is accelerated.

number6.9 A figure skater spins counterclockwise, so her angular velocity is generally taken to be positive. (Luu, Wikimedia Commons)

The relationship between the size oftangential acceleration,INand angular acceleration,

$IN,YIN=rINoIN= IN r . IN,YIN=rINoIN= IN r .$

6.10

These equations imply that the magnitudes of the tangential and angular accelerations are proportional to each other. The greater the angular acceleration, the greater the change in tangential acceleration, and vice versa. For example, consider a passenger resting on a Ferris wheel. A Ferris wheel with higher angular acceleration will give the rider more tangential acceleration because as the Ferris wheel increases its rotation rate, its rotation rate also increases.tangential speedNote that the radius of the rotating object is also important. For example, for a given angular acceleration$IN IN$, a smaller Ferris wheel will bring less tangential acceleration to the rider.

### The secret of success

Tangential acceleration is sometimes expressed asINton.It's alinear accelerationAlong a direction tangent to the circle at a point of interest for a circular or rotary movement. Remember that the tangential acceleration is parallel (equal or opposite) to the tangential velocity.centripetal accelerationAlways perpendicular to the tangential velocity.

So far we have defined three rotation variables:$I I$,$Oh Oh$, y$IN IN$These are the angular versions of the linear variables X, V, and A. The following equations in the table represent the magnitude of the rotation variable and only when the radius is constant and perpendicular to the rotation variable.Table 6.2Show how they are related.

rotary lineal relationship
$I I$ X $I= X r I= X r$
$Oh Oh$ v $Oh= v r Oh= v r$
$IN IN$ IN $IN= IN r IN= IN r$

board 6.2 Rotational Variables and Linear Variables

Now we can start to see what the rotations look like.$I I$,$Oh Oh$, y$IN IN$related to each other. For example, if a motorcycle wheel starts from rest with a large angular acceleration and continues for a long time, it will end up spinning rapidly and revving a lot. From a variable point of view, if the angular acceleration of the wheel$IN IN$great for a long timeton, then the final angular velocity$Oh Oh$and angle of rotation$I I$very big. In the case of linear motion, if an object starts from rest and experiences a large linear acceleration, it has a large terminal velocity and will travel a large distance.

This hereRotation movement kinematicsDescribe the relationship between angle of rotation, angular velocity, angular acceleration, and time. it's justdescribeMotion: Does not include any force or mass that might affect rotation (these are part of dynamics). Remember the kinematic equations for linear motion:$v= v 0 +INton v= v 0 +INton$(consistentIN).

As in linear kinematics, we assumeINis a constant, which means that the angular acceleration$IN IN$is also a constant because$IN=rIN IN=rIN$The kinematic relationship equation between$Oh Oh$,$IN IN$, ytonY

$Oh= Oh 0 +INton(consistentIN), Oh= Oh 0 +INton(consistentIN),$

Where$Oh 0 Oh 0$is the initial angular velocity. Note that the equations are the same as in the linear version, except for the angular simulation of the linear variable. In fact, all linear kinematic equations have rotational analogues, iTable 6.3.These equations can be used to solve rotational or linear kinematic problems whereINy$IN IN$it's constant

rotary lineal
$I= Oh ¯ ton I= Oh ¯ ton$ $X= v ¯ ton X= v ¯ ton$
$Oh= Oh 0 +INton Oh= Oh 0 +INton$ $v= v 0 +INton v= v 0 +INton$ consistent$IN IN$,IN
$I= Oh 0 ton+ 1 2 IN ton 2 I= Oh 0 ton+ 1 2 IN ton 2$ $X= v 0 ton+ 1 2 IN ton 2 X= v 0 ton+ 1 2 IN ton 2$ consistent$IN IN$,IN
$Oh 2 = Oh 0 2 +2INI Oh 2 = Oh 0 2 +2INI$ $v 2 = v 0 2 +2INX v 2 = v 0 2 +2INX$ consistent$IN IN$,IN

board 6.3 Rotational kinematics equations

In these equations,$Oh 0 Oh 0$y$v 0 v 0$is the initial value,$ton 0 ton 0$is zero, the mean angular velocity$Oh ¯ Oh ¯$and average speed$v ¯ v ¯$Y

$Oh ¯ = Oh 0 +Oh 2 y v ¯ = v 0 +v 2 . Oh ¯ = Oh 0 +Oh 2 y v ¯ = v 0 +v 2 .$

6.11

### interesting physics

#### chasing the wind

number6.10 The tornado descends from the clouds in a violent spinning motion. (Daphne Zaras, NOAA)

Storm chasers tend to fall into one of three categories: hobbyists who chase tornadoes as a hobby, atmospheric scientists who collect data for research, weather observers for the media, or scientists who wear their work as a disguise for fun. Storm chasing is a dangerous hobby because tornadoes can change course quickly without warning. It is common for a flat tire to be replaced due to debris left on the road when storm chasers track tornado damage. The most active region for tornadoes in the world, known astornado alley, located in the central United States, between the Rocky Mountains and the Appalachian Mountains.

A tornado is a perfect example of spinning motion in nature. They come from intense thunderstorms called supercells, where a column of air swirls around a horizontal axis, usually about four miles wide. The difference in wind speed between the strong cold winds higher in the atmosphere in the jet stream and the weaker winds blowing north from the Gulf of Mexico causes the axis of the rotating column of air to move with the movement of the storm, which causes the axis to become vertical, creating a tornado.

Tornadoes produce wind speeds of up to 500 km/h (around 300 mph), especially at the narrower base of the funnel, as rotation speed increases with decreasing radius. They blow up houses as if they were made of paper and have been known to poke slivers of straw through logs.

### torque

If you've ever spun a bicycle wheel or pushed a merry-go-round, you know that force is needed to change angular velocity. The longer the force is appliedcentral point(osupport), the greater the angular acceleration. For example, if you are too close to the hinge, the door will open slowly, but if you are too far from the hinge, the door will swing open. We also know that the bigger the door, the slower it opens; this is because angular acceleration is inversely proportional to mass. These relationships are very similar to those between force, mass, and acceleration in Newton's second law of motion. Since we've covered the angular versions of distance, velocity, and time, you might be wondering what the angular version of force is and how it relates to linear forces.

The angular version of the power istorque $ton ton$, which is the power steering efficiency. seeFigure 6.11.The equation for the magnitude of the torque is

$ton=rFcrimeI, ton=rFcrimeI,$

Whereryes rangelever,Fis the magnitude of the linear force, and$II$is the angle between the moment arm and the force. The weight arm is the vector from the point of rotation (pivot or fulcrum) to where the force is applied. Since the magnitude of the lever arm is distance, its unit is the meter, and the unit of torque is N·m. Momentum is a vector and has the same direction as the angular acceleration it produces.

number6.11 A person at the edge of the carousel pushes the carousel perpendicular to the lever arm to achieve maximum torque.

The application of a stronger torque will give a greater angular acceleration. Like the harder a man pushes a merry-go-roundFigure 6.11, the faster it accelerates. The bigger the carousel, the slower it will accelerate for the same amount of torque. If the person wants to maximize the effect of his force on the merry-go-round, he should push as far from the center as possible to achieve the greatest lever arm and thus the greatest torque and angular acceleration. Torque is also maximized when force is applied perpendicular to the lever arm.

### teacher support

#### teacher support

[knowledge][OL][Alabama]Demonstrates the physical relationship between torque, force, the angle at which the force is applied, and the length of the bar arm using handles of different lengths. Help students make connections between physical observations and mathematical relationships. For example, the torque is greatest when the force is applied completely perpendicular to the lever arm due to$crimeI=1 crimeI=1$a$I=after 90 I=after 90$Every time.

### Solve rotational kinematics and torque problems

Just as linear forces can balance to produce zero net force and no linear acceleration, so can rotational motion. When two couples of the same magnitude act in opposite directions, there is no net torque and no angular acceleration, as you can see in the video below. If zero net torque is applied to a system rotating at a constant angular velocity, the system will continue to rotate at the same angular velocity.

### look at the physics

#### introduction to the pair

This herevideoThe moment is defined as the moment arm (same as the lift arm). It also covers the topic of forces acting in opposite directions around a pivot point. (At this stage, you can ignore the labor and mechanical advantages Sal mentioned.)

Click to see the content

If the net torque acting on the scale in the example is positive instead of zero, how does this affect the angular acceleration? What happens to those in power over time?

1. The ruler is in rotational equilibrium, so it does not rotate about its center of mass. Therefore, the angular acceleration will be zero.

2. The ruler is not in rotational equilibrium, so it does not rotate about its center of mass. Therefore, the angular acceleration will be zero.

3. The ruler is not in rotational equilibrium, so it rotates about its center of mass. Therefore, the angular acceleration will not be zero.

4. The ruler is in rotational equilibrium, so it will rotate about its center of mass. Therefore, the angular acceleration will not be zero.

Now let's look at an example of applying rotational kinematics to a fishing reel and the concept of torque to a merry-go-round.

### Example

#### Calculate when the reel stops spinning

A deep-sea fisherman uses a rod with a spool radius of 4.50 cm. A large fish took the bait and swam away from the boat, pulling the line from its reel. When the fishing line is unwound from the spool, the spool rotates at an angular velocity of 220 rad/s. A fisherman brakes a spinning wheel and produces an angular acceleration of -300 rad/s2How long does it take for the roll to stop? ### strategy

they asked us to find out whentonStop the roll. The magnitude of the initial angular velocity is$Oh 0 =220 Oh 0 =220$rad/s, and the magnitude of the final angular velocity$Oh=0 Oh=0$.The sign magnitude of the angular acceleration is$IN=−300 IN=−300$rad/s2, where the minus sign indicates that it acts in the opposite direction to the angular velocity. If we look at the rotational kinematics equations, we see all quantities excepttonknown in the equation$Oh= Oh 0 +INton Oh= Oh 0 +INton$, so it is the simplest equation to solve this problem.

solution

The equation used is$Oh= Oh 0 +INton Oh= Oh 0 +INton$.

We solve the equation algebraically astonand insert known values.

$ton = Oh− Oh 0 IN = 0−220rad/s −300 rad/s 2 = 0,733Other ton = Oh− Oh 0 IN = 0−220rad/s −300 rad/s 2 = 0,733Other$

6.12

to talk

The time to stop the roll is relatively short due to the significant acceleration. Line sometimes breaks due to force, and anglers often let the fish swim for a while before slowing down the reel. A tired fish will slow down and require less acceleration and therefore less power.

### Example

#### Calculation of the carousel torque.

consider the person who pushes the fairground carouselFigure 6.11.He exerts a force of 250 N on the edge of the merry-go-round, perpendicular to a radius of 1.50 m, how much torque does he produce? Assume that the friction acting on the merry-go-round is negligible.

### strategy

To find the torque, note that the applied force is perpendicular to the radius and that friction is negligible.

solution

$ton=rFcrimeI=(1,50risa)(250No)crime(Pi2).=375No⋅risaton=rFcrimeI=(1,50risa)(250No)crime(Pi2).=375No⋅risa$

6.13

to talk

A person maximizes torque by applying a force perpendicular to the lever arm, so$I= Pi 2 I= Pi 2$y$crimeI=1 crimeI=1$The guy also maximizes his torque by pushing the outer edge of the carousel so that he can get the biggest lever arm possible.

### practice questions

15.

If a person applies a moment, how much torque does he produce?12\,\text{N}fortaleza1.0\,\text{m}Away from the fulcrum, perpendicular to the lever arm?

1. \frac{1}{144}\,\text{N-m}

2. \frac{1}{12}\,\text{N-m}

3. 12\,\tekst{N-m}

4. 144\,\tekst{N-m}

sixteen.

The angular velocity of the object changes from 3 rad/s clockwise to 8 rad/s clockwise in 5 seconds. What is its angular acceleration?

17.

What is angular acceleration?

1. Angular acceleration is the rate of change of angular displacement.

2. Angular acceleration is the rate of change of angular velocity.

3. Angular acceleration is the rate of change of linear displacement.

4. Angular acceleration is the rate of change of linear velocity.

18.

What is the equation for angular acceleration,IN? thinkIis the angle,Ohis the angular velocity, andtonis the time.

1. $IN= commandOh commandton IN= commandOh commandton$
2. $IN=commandOhcommandton IN=commandOhcommandton$
3. $IN= commandI commandton IN= commandI commandton$
4. $IN=commandIcommandton IN=commandIcommandton$

19.

Which of the following best describes torque?

1. It is the rotational equivalent of the force.

2. It is the force that affects linear motion.

3. It is the rotational equivalent of acceleration.

4. It is acceleration that affects linear motion.

20.

What is the equation for torque?

1. \tau = {F\,cos\theta}\,{r}

2. \tau = \frac{F\sin\theta}{r}

3. \tau = rF\!\cos \theta

4. \tau = rF\!\sin \theta

### teacher support

#### teacher support

Use the Check Your Understanding questions to assess whether students have mastered the learning objectives in this section. If students are having trouble with a particular objective, these questions will help identify which objective is causing the problem and direct students to the appropriate content.

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