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The given figure shows the top view of an old fashioned vinyl record. Two perpendicular line segments are drawn through the center of the circular record, one vertically upward and one horizontal to the right side. Two flies are shown at the end points of the vertical lines near the borders of the record. Two arrows are also drawn perpendicularly rightward through the end points of these vertical lines depicting linear velocities. A curved arrow is also drawn at the center circular part of the record which shows the angular velocity.
As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case.

Phet explorations: ladybug revolution

Ladybug Revolution

Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs.

Section summary

  • Uniform circular motion is motion in a circle at constant speed. The rotation angle Δ θ size 12{Δθ} {} is defined as the ratio of the arc length to the radius of curvature:
    Δ θ = Δ s r , size 12{Δθ= { {Δs} over {r} } ","} {}

    where arc length Δ s size 12{Δs} {} is distance traveled along a circular path and r size 12{r} {} is the radius of curvature of the circular path. The quantity Δ θ size 12{Δθ} {} is measured in units of radians (rad), for which

    rad = 360º 1  revolution. size 12{2π`"rad"=`"360""°="`1`"revolution."} {}
  • The conversion between radians and degrees is 1 rad = 57 . 3 º size 12{1"rad"="57" "." 3°} {} .
  • Angular velocity ω size 12{ω} {} is the rate of change of an angle,
    ω = Δ θ Δ t , size 12{ω= { {Δθ} over {Δt} } ","} {}

    where a rotation Δ θ size 12{Δθ} {} takes place in a time Δ t size 12{Δt} {} . The units of angular velocity are radians per second (rad/s). Linear velocity v size 12{v} {} and angular velocity ω size 12{ω} {} are related by

    v =  or  ω = v r . size 12{v=rω``"or "ω= { {v} over {r} } "."} {}

Conceptual questions

There is an analogy between rotational and linear physical quantities. What rotational quantities are analogous to distance and velocity?

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Problem exercises

Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions—it then calculates the distance traveled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometers should the odometer read?

723 km

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Microwave ovens rotate at a rate of about 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second?

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An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?

5 × 10 7 rotations size 12{9 "." "79" times "10" rSup { size 8{8} } `"rotations"} {}

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(a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of 6 . 4 × 10 6 m size 12{6 "." 4 times "10" rSup { size 8{6} } } {} at its equator, what is the linear velocity at Earth’s surface?

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A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher’s hand is 35.0 m/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?

117 rad/s

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In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball?

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A truck with 0.420-m-radius tires travels at 32.0 m/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev/min?

76.2 rad/s

728 rpm

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Integrated Concepts When kicking a football, the kicker rotates his leg about the hip joint.

(a) If the velocity of the tip of the kicker’s shoe is 35.0 m/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip’s angular velocity?

(b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m/s?

(c) Find the maximum range of the football, neglecting air resistance.

(a) 33.3 rad/s

(b) 500 N

(c) 40.8 m

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Construct Your Own Problem

Consider an amusement park ride in which participants are rotated about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a free body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders’ clothing and the wall.

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Questions & Answers

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
Collins Reply
What is thermal heat all about
Abel Reply
why uniform circular motion is called a periodic motion?.
Boniface Reply
when a train start from A & it returns at same station A . what is its acceleration?
Mwdan Reply
what is distance of A to B of the stations and what is the time taken to reach B from A
BELLO
the information provided is not enough
aliyu
Hmmmm maybe the question is logical
yusuf
where are the parameters for calculation
HENRY
there is enough information to calculate an AVERAGE acceleration
Kwok
mistake, there is enough information to calculate an average velocity
Kwok
~\
Abel
what is the unit of momentum
Abel
wha are the types of radioactivity ?
Worku Reply
what are the types of radioactivity
Worku
what is static friction
Golu Reply
It is the opposite of kinetic friction
Mark
static fiction is friction between two surfaces in contact an none of sliding over on another, while Kinetic friction is friction between sliding surfaces in contact.
MINDERIUM
I don't get it,if it's static then there will be no friction.
author
It means that static friction is that friction that most be overcome before a body can move
kingsley
static friction is a force that keeps an object from moving, and it's the opposite of kinetic friction.
author
It is a force a body must overcome in order for the body to move.
Eboh
If a particle accelerator explodes what happens
Eboh
why we see the edge effect in case of the field lines of capacitor?
Arnab
what is wave
Muhammed Reply
what is force
Muhammed
force is something which is responsible for the object to change its position
MINDERIUM
more technically it is the product of mass of an object and Acceleration produced in it
MINDERIUM
wave is disturbance in any medium
iqra
energy is distributed in any medium through particles of medium.
iqra
If a particle accelerator explodes what happens
Eboh Reply
we have to first figure out .... wats a particle accelerator first
Teh
What is surface tension
Subi Reply
The resistive force of surface.
iqra
Who can tutor me on simple harmonic motion
yusuf Reply
on both a string and peldulum?
Anya
spring*
Anya
Yea
yusuf
Do you have a chit-chat contact
yusuf
I dont have social media but i do have an email?
Anya
Which is
yusuf
Where are you chatting from
yusuf
I don't understand the basics of this group
Jimmy
teach him SHM init
Anya
Simple harmonic motion
yusuf
how.an.equipotential.line is two dimension and equipotential surface is three dimension ?
syed Reply
definition of mass of conversion
umezurike Reply
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
Marisa
how many subject is in physics
Adeshina Reply
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
Praise what level are you
yusuf
If u are doing a levels in your first year you do AS topics therefore you do 5 big topic i.e particles radiation, waves and optics, mechanics,materials, electricity. After that you do A level topics like Specific Harmonic motion circular motion astrophysics depends really
Anya
Yeah basics of physics prin8
yusuf
Heat nd Co for a level
yusuf
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
Tamdy Reply
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
hallow
Boniface
please send the answer
Boniface
the range of objects and phenomena studied in physics is
Bethel Reply
I don't know please give the answer
Boniface
Practice Key Terms 6

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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