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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

what is torque
Deepak Reply
The turning effect of force is called torque.
what there factors affect the surface tension of a liquid
Promise Reply
formula for impedance
muyiwa Reply
ehat is central forces
Nita Reply
what is distance?
Jonathan Reply
What does mean ohms law imply
Victoria Reply
ohms law state that the electricity passing through a metallic conductor is directly proportional to the potential difference across its end
what is matter
folajin Reply
Anything that occupies space
Any thing that has weight and occupies space
Anything which we can feel by any of our 5 sense organs
what is a sulphate
any answers
the time rate of increase in velocity is called
Blessing Reply
What is uniform velocity
Greetings,users of that wonderful app.
Frank Reply
how to solve pressure?
Cruz Reply
how do we calculate weight and eara eg an elefant that weight 2000kg has four fits or legs search of surface eara is 0.1m2(1metre square) incontact with the ground=10m2(g =10m2)
can someone derive the formula a little bit deeper?
what is coplanar force?
forces acting and lying on d same plane
what is accuracy and precision
Peace Reply
How does a current follow?
Vineeta Reply
which one dc or ac current.
how does a current following?
AC current
AC current follows due to changing electric field and magnetic field.
you guys are just saying follow is flow not follow please
ok bro thanks
but i wanted to understand him/her in his own language
but I think the statement is written in English not any other language
my mean that in which form he/she written this,will understand better in this form, i write.
ok thanks bro. my mistake
u are welcome
what is a semiconductor
Vineeta Reply
substances having lower forbidden gap between valence band and conduction band
what is a conductor?
replace lower by higher only
convert 56°c to kelvin
How does a current follow?
A semiconductor is any material whose conduction lies between that of a conductor and an insulator.
what is Atom? what is molecules? what is ions?
Abubakar Reply
atoms are the smallest unit of an element which is capable of behaving as a single unit
a molecule is d smallest unit of a substances capable of independent existence and can also retain the chemical proper ties of that substance
an ion is referred to as freely moving charged particles
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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