Rotational Motion

Chapter 7

Measuring Rotational MotionWhen an object spins it is said to

undergo rotational motion.The axis of rotation is the line about

which the rotation occursA point on an object that rotates

about a single axis undergoes circular motion around that axisAny given point on an object of any shape rotates about the axis in a circular pattern

Measuring Rotational MotionDifficult to describe the motion of

an object in rotational motion using only information for the linear caseThis is because the path is consistently changing

When rotational motion is described using angles, all points on a rigid body move through the same angle in the same interval of time

Measuring Rotational MotionTo analyze rotational motion, choose a

fixed reference lineThe distance a point moves is called the arc length and is denoted by the variable s

Reference Line

r Reference Line

r

s

Measuring Rotational MotionAngles can be measured in radiansPrior to this point, we have used

degreesA radian is an angle whose arc

length is equal to its radius, which is approximately equal to 57.3

Be sure to convert to radians in this chapter!!

Defined as an equation: r

s

Measuring Rotational MotionBecause radius is defined as arc length

divided by radius, both are distancesThe units are canceled and the

abbreviation rad is used in their place360 is equal to 2 radThe arc length s is equal to the

circumference of a circle, if the object travels the full rotation

The circumference of a circle is 2r

Measuring Rotational MotionTo convert from degrees to radians:

(deg)180

)( rad

Measuring Rotational MotionAngular displacement describes

how much an object has rotatedThe angular displacement traveled

is equal to the change in arc length divided by the radius from the axis of rotation to that point

axis from Distance

length arcin Changeradians)(in nt DisplacemeAngular

r

s

Measuring Rotational MotionIn general, a positive arc length is

taken to be rotating counter clockwise and a negative arc length rotates clockwise

Measuring Rotational MotionPractice Problem 1Earth has an equatorial radius of

approximately 6380 km and rotates 360 in 24h.What is the angular displacement, in degrees, of a person standing on the equator for 1.0h?

Covert this angular displacement to radians.

What is the arc length traveled by this person?

Measuring Rotational MotionAngular speed describes the rate

of rotationThe average angular speed of a

rotating rigid object is the ration of the angular displacement to the time interval it takes for the object to undergo that displacement

It describes how quickly the rotation occurs

Interval Time

ntDisplacemeAngular SpeedAngular Average

tavg

Measuring Rotational MotionAngular speed is given in units of

radians per second (rad/s)Sometimes given in revolutions per

unit timeRemember that 1 revolution is 2rad

Measuring Rotational MotionPractice Problem 2Before the advent of compact

discs, musical recordings were commonly sold on vinyl discs that could be played on a turntable at 45 rpm (revolution per minute) or 33.3 rpm. Calculate the corresponding angular speeds in rad/s.

Measuring Rotational MotionPractice Problem 3An Indy car can complete 120 laps

in 1.5h. Even though the track is oval rather than a circle, you can still find the average angular speed. Calculate the average angular speed of the Indy car in rad/s.

Measuring Rotational MotionAngular acceleration occurs

when angular speed changesThe average angular acceleration is given below.

The units for angular acceleration are rad/s2

Interval Time

SpeedAngular in Change on AcceleratiAngular Average

12

12

tttavg

Measuring Rotational MotionPractice Problem 4A yo-yo at rest is sent spinning at

an angular speed of 12 rev/s in 0.25 s. What is the average angular acceleration of the yo-yo?

Measuring Rotational MotionPractice Problem 5A top spinning at 15 rev/s spins for

55 s before coming to a stop. What is the average angular acceleration of the top while it is slowing?

Measuring Rotational MotionPractice Problem 6A certain top will remain stable

(upright) at angular speeds above 3.5 rev/s. The top slows due to friction at a rate of 1.3 rad/s2. What initial speed (in rad/s and rev/s) must the top be given in order to spin for at least 1 min?

Measuring Rotational MotionAll points on a rotating rigid object

have the same angular acceleration and angular speed

For a rotating object to remain rigid every portion of the object must have the same angular speed and the same angular acceleration, if not then the object changes shape

Measuring Rotational MotionCompare the angular motion

equations to those we already know for linear motion

The equations are similar, however the rotational equations replace linear variables with their rotational counterparts

a

v

x

AngularLinear

Linear and Rotational Kinematic Equations

Linear Motion with Constant Acceleration

Rotational Motion is Constant Angular Acceleration

tif atvv if 222

ifadvv

if222

)(2

1 2tti )(2

1 2tatvd i

tfi

2

)(

t

vvd fi

2

)(

Measuring Rotational MotionPractice Problem 7A barrel is given a downhill rolling start of

1.5 rad/s at the top of a hill. Assume a constant angular acceleration of 2.9 rad/s2.If it takes 11.5 s to get to the bottom of the hill, what is the final angular speed of the barrel?

What angular displacement does the barrel experience during the 11.5 s ride?

Tangential and Centripetal Acceleration

Sometimes it is useful to describe the motion of a rotating object in terms of linear speed and linear acceleration of a single point on that object

Objects in circular motion have a tangential speed

Tangential speed is the instantaneous linear speed of an object directed along the tangent to the objects circular path

Tangential and Centripetal AccelerationTangential speed is also known as the

instantaneous linear speed of that point

The tangential speed of two different points at different distances from the center have different tangential speeds

If the angular displacement is the same and the radius increases, the arc length traveled must also increase. Therefore the point on the outside has a larger tangential speed

Tangential and Centripetal AccelerationTo find tangential speed:

Note that is the instantaneous angular speed, not the average angular speed

This is only valid when is measured in radian

SpeedAngular Radius Speed Tangential rvt

Tangential and Centripetal AccelerationPractice Problem 8A golfer has a maximum angular

speed of 6.3 rad/s for her swing. She can choose between two drivers, one placing the club head 1.9 m from her axis or rotation and the other placing it at 1.7 m from the axisFind the tangential speed of the club head for each driver.

All other factors being equal, which driver is likely to hit the ball farther?

Tangential and Centripetal AccelerationTangential acceleration is tangent

to the circular pathThe tangential acceleration is

the instantaneous linear acceleration of an object directed along the tangent to the objects circular path

To find the tangential acceleration:onAcceleratiAngular Radius on Accelerati Tangential

rat

Tangential and Centripetal AccelerationPractice Problem 9A yo-yo has a tangential

acceleration of 0.98 m/s2 when it is released. The string is wound around the central shaft of radius 0.35 cm. What is the angular acceleration of the yo-yo?

Tangential and Centripetal AccelerationIf you are moving around a circle with a

constant tangential speed, you are still accelerating

This is because you are changing direction

The acceleration caused by the change in direction is called centripetal acceleration

Centripetal acceleration is an acceleration directed toward the center of a circular path

Tangential and Centripetal AccelerationTo find the centripetal

acceleration:

2

2

2

2

SpeedAngular Radius on Accelerati lCentripita

Radius

Velocity Tangential on Accelerati lCentripeta

ra

r

va

c

tc

Tangential and Centripetal AccelerationDO NOT replace centripetal with

centrifugal, centripetal mean center seeking and centrifugal means center fleeing. They are opposite in meaning!!!

Tangential and Centripetal AccelerationPractice Problem 10A cylindrical space station with a 115

m radius rotates around its longitudinal axis at an angular speed of 0.292 rad/s. Calculate the centripetal acceleration on a person at the following locationsAt the center of the stationHalfway to the rim of the stationAt the rim of the station

Tangential and Centripetal AccelerationTangential and Centripetal accelerations

are perpendicular to each otherTangential acceleration is due to

changing speedCentripetal acceleration is due to

changing directionYou may find the magnitude of the total

acceleration by using Pythagoreans theorem

You can also find the direction by using the inverse tangent function

Causes of Circular MotionThe inertia of an object tends to

maintain the object’s motion in a straight line path

Circular motion is possible because of the force that is directed towards the axis of rotation

This force can be found by applying Newton’s Second Law in the radial direction

Causes of Circular MotionForce that maintains circular

motion can be found by:

2

2

2

2

SpeedAngular Radius Mass Force

Radius

Speed Tangentialmass Force

mrF

r

mvF

c

tc

Causes of Circular MotionThe force needed to maintain

circular motion is no different than any other force we have discussed

An example of this can be seen where the tires of a car encounter friction in order to make it move in a circular path

Causes of Circular MotionPractice Problem 11An astronaut who weighs 735 N on

Earth is at the rim of a cylindrical space station with a 73 m radius. The space station is rotating at an angular speed of 3.5 rpm. Evaluate the force that maintains the circular motion of the astronaut.

Causes of Circular MotionA force directed toward the center

is necessary for circular motionIf this force vanishes, the object

does not continue to move in a circular path, but it continues in a straight line path that is tangent to the circular path it was in

Causes of Circular MotionOnce this force vanishes, it continues

in motion as we have studied previously

For example, if a ball was attached to a string being swung vertically in a circle and the string broke at the top of its path, the ball would continue as a projectile launched horizontally and you can continue to solve the problem as before.

Causes of Circular MotionDescribe what happens when a car makes a fast turn.

What causes the passenger to move toward the door?INERTIAThe passenger is originally moving in a straight line path

When the car makes the turn, the passenger wants to continue in the straight line path until an outside force create a change in the direction of the person

The force of the door on the person is what makes the person turn and follow the circular path