Rotary Motion

Physics

Montwood High School

• Rotary motion is the motion of a body around an internal axis.– Rotary motion – axis of rotation is inside the

moving object. Ex. Spinning wheel.– Circular motion – axis of motion is outside the

moving object. Ex. Object on the rim of the spinning wheel.

• The angles in rotary motion are measured in radians; linear motion and circular motion are measured in degrees.– Calculator will have to be in radian mode

(indicated by rad on the screen)

Radian Definition

• When the arc length S is equal to the length of the radius r, the angle swept out by r is 1 radian.

• Any angle measured in radians is defined by the following:

rs

Radian Definition• Angular displacement

– how much of a circle the object moves through.

• Because S and r are measured in meters, the units cancel and the unit rad is added (the What Happens to the Radians article will have you substitute rad/m).

360 degrees = 2 radians = 1 revolution (or rotation)

Conversions• Degrees to Radians

example:• Radians to Degrees

example:

rad0472.1180

rad60

rad0472.1360

rad260

oo

oo

o

o

o

o

94.85

rad180

rad5.1

94.85

rad2360

rad5.1

Conversions• Revolutions to

Radians example:

• Radians to Revolutions example:

rad85.18

rev1rad2

revs3

rev

radrev

rad

6366.0

21

4

Angular Displacement • Angular displacement

describes how much an object has rotated.

• Counterclockwise displacements are considered positive (as shown).

• Clockwise rotations are considered negative.

mrad

,rad:unit

rS

Angular Speed • Angular speed is

the speed at which the object is rotating.

• Unit: rad/s; rev/s; etc.

t

Angular Speed

• Angular velocity is a vector quantity.

• To determine the direction of the angular velocity vector, curl fingers of right hand in the direction of rotation. Thumb points in the direction of the angular velocity vector .

Angular Speed

Angular Acceleration • Angular acceleration is the rate at which

the rotational speed changes.

• Unit: rad/s2; rev/s2; etc.

• Increasing rate of rotation

f > i, positive

• Decreasing rate of rotation

f < i, negative

ttif

Angular Speed Revisited• For a rotating object,

every point on the object has the same angular velocity and the same angular acceleration .

• The tangential (linear) velocity v or vT of any point is proportional to its distance from the axis of rotation.

• The linear distance d is equal to the arc length s of the rotating object.

rv

Angular Speed Revisited

• If a point on the edge of a rotating object had an angular speed greater than that of a point near the center, the shape of the object would be changing.

• For a rotating object to remain rigid, every point of the object must have the same angular speed and angular acceleration.

Angular Speed Revisited

• All skaters have the same angular velocity and the same angular acceleration , however, the linear or tangential velocity vT increases from the center along the radius to the last person.

Linear vs. Angular Quantities

Linear vs. Angular Quantities

• Time = t• Angular

acceleration = • Initial angular

velocity = i• Final angular

velocity = f• Angular

displacement = • Radius = r

2

t

t5.0t

2t

2i

2f

if

2i

fi

Transferring Rotational Motion• If two disks are touching

each other as in the figure, the top disk is driven by a motor and turns the 2nd disk (the wheel) by making use of the friction between them.

• The relationship between the diameters D of the two disks and the number N of revolutions is:

2211NDND

Transferring Rotational Motion

• Using two disks to transfer rotational motion is not very efficient due to slippage that may occur between disks. The most common way to prevent disk slippage:– place teeth on the edge of the disk (b).– Connect the disks with a belt (c).

• Instead of using disks, we use gears or belt-driven pulleys to transfer the rotational motion.

• Teeth on the gears eliminate slippage and provides for distance between rotating centers.

Transferring Rotational Motion

Transferring Rotational Motion

• Rearrange to get

• D1/D2 is the ratio of the diameters of the disks. If the ratio is 2, this means that D1 has a diameter that is 2 times greater than D2.

• N2/N1 is the ratio of the number of revolutions of the two disks. If the ratio is 2, the smaller disk makes two revolutions while the larger disk makes one revolution.

2211NDND

1

2

2

1

N

N

D

D

Transferring Rotational Motion• Pulleys and gears are used to increase or decrease

the angular velocity of a rotating shaft or wheel.• When two gears or pulleys are connected, the

speed at which each turns compared to the other is inversely proportional to the diameter of that gear or pulley.– The larger the diameter of a pulley or gear, the slower it

turns.– The smaller the diameter of the pulley or gear, the faster

it turns when connected to the larger one.– http://www.technologystudent.com/gears1/pulley2.htm– http://www.technologystudent.com/gears1/pulley3.htm

Pulleys, Chains, Gears, Etc.• The two wheels have different

radii and will turn at different angular speeds.

• The linear velocity v of the belt is constant. One part of the belt cannot travel faster than any other part of the belt.

• The linear velocity v where the belt contacts the pulley wheel must be the same for each pulley wheel in order to turn both wheels.

• v1 = v2, so r1·w1 = r2 · w2

Problem Example On the runway waiting

for take-off, the blades in a jet engine spin with an angular velocity of 110 radians per second. During take-off, the blades accelerate to an angular velocity of 330 rad/s in just 14 seconds. What was the angular acceleration of the blades?

f i

t

=rad / s rad / s

14s= rad / s2

330 110

16

Problem Example The blades of a blender

spin with an angular velocity of 375 rad/s when in “blend” mode. Pushing the “puree” button accelerates the blades with an angular acceleration of 1740 rad/s2. What is the angular velocity of the blades after 7 revs (44 rad)?

Problem Example

srad

98.541

rad44s

rad17402

srad

375

2

2

f

2

2

f

2if

2i

2f

• A device called a stroboscope or strobe light may be used to measure or check the speed of rotation of a shaft or other machinery part.

• The stroboscope is used to slow down repeating motion to be observed more conveniently.

• The light flashes rapidly and the rate of the flash can be adjusted to coincide with the rotation of a point or points on the rotating object.

• Knowing the rate of flashing will also then reveal the rate of rotation.

• A slight variation in the rate of rotation and the flash will cause the observed point to appear to move either forward or backward.

• http://en.wikipedia.org/wiki/File:Strobe.gif• Consider the stroboscope as used in mechanical analysis.

This may be a “strobe light” that is fired at an adjustable rate. Suppose you are looking at something rotating at 60 revolutions per second: if you view it with a series of short flashes at 60 times per second, each flash illuminates the object at the same position in its rotational cycle, so it appears that the object is stationary. Furthermore, at a frequency of 60 flashes per second, persistence of vision smooths out the sequence of flashes so that the perceived image is continuous.

• If you view the same rotating object at 61 flashes per second, each flash will illuminate it at a slightly earlier part of its rotational cycle. Sixty-one flashes will occur before you see the object in the same position again, and you will perceive the series of images as if it is rotating backwards once per second.

• The same effect occurs if you view the object at 59 flashes per second, except that each flash illuminates it a little later in its rotational cycle and so, it seems to be slowly rotating forwards.

• In the case of motion pictures, action is captured as a rapid series of still images and the same stroboscopic effect can occur.

• Wagon-wheel effect: Motion-picture cameras conventionally film at 24 frames per second. Although the wheels of a vehicle are not likely to be turning at 24 revolutions per second (as that would be extremely fast), suppose each wheel has twelve spokes and rotates at only two revolutions per second. Filmed at 24 frames per second, the spokes in each frame will appear in exactly the same position. Hence, the wheel will be perceived to be stationary. In fact, each photographically captured spoke in any one position will be a different actual spoke in each successive frame, but since the spokes are close to identical in shape and color, no difference will be perceived.

• If the wheel rotates a little more slowly than two revolutions per second, the position of the spokes is seen to fall a little further behind in each successive frame and therefore the wheel will seem to be turning backwards.

• If the wheel rotates a little more slowly than two revolutions per second, the position of the spokes is seen to fall a little further behind in each successive frame and therefore the wheel will seem to be turning backwards.