lecture 9 chapter 8 rotational motion - sfsu physics &...

Physics 10114-Sep-10 Physics 101

Chapter 8Rotational Motion

Lecture 9

Quiz -- Monday Sept. 20 (Chaps. 4-7)

Physics 101

Announcements

• My office hour today was 1010-1100 instead of 1310-1400.

• My help session today is 1510-1600 in TH118.

• Quiz Monday

Physics 101

Rotational MotionIn physics we distinguish two types of motion

for objects:• Translational Motion (change of location).

Whole object moves through space.• Rotational Motion - object turns around an

axis (axle); axis does not move. (Wheels)We’ve mostly discussed translational motion;

today we consider rotation.

Physics 101

Rotational Motion• Study rotational motion of solid object with

fixed axis of rotation (axle)• Have “angular” versions of the quantities

we studied in translational motion -position, distance, velocity, acceleration

• Later look at angular versions of force, mass, momentum, and kinetic energy.

• Use Greek letters for most angular quantities

Physics 101

Angular PositionAngular Position

We will measure angular position in revolutions

Counterclockwise (CCW): positive rotation

Clockwise (CW): Negative rotation

Physics 10114-Sep-10

Circles

Circumference is distance around the perimeter of a circle.

For circle of radius R, circumference is 2πR.

Axis

Radius

Diameter

If point on circle makes one revolution, it travels a distance 2πR.

•

Physics 101

Translational (Tangential) Distance d vs. Angular Distance ∆θ

Translational (Tangential) Distance d vs. Angular Distance ∆θ

∆θ

For a point at radius R on the wheel, d = 2πR∆θ R

for ∆θ in revolutions

Physics 101

(Average) Angular Velocity ω(Average) Angular Velocity ωAvg. Angular Velocity = # Revolutions/(Time Taken)

ω = ∆θ / t

Unit: Revolutions/s or Revolutions/min (RPM)

Sign convention : ω is positive for counterclockwise rotation, negative for clockwise rotation

Example: Wheel turns CCW through 60 revolutions in 2 minutes. What is the rotational speed?

ω = +60 Rev/(2 min) = 30 RPM or 0.5 rev/s

Physics 101

Translational vs. Rotational SpeedEvery point on a rotating body has an angular velocity ω and a tangential velocity vt. vt = d/∆t = 2πR(∆θ)/∆t = 2πRω

vt

R

d


Speed in Circular MotionRotational Speed ω: Revolutions/s or RPMTangential Speed vt: Total distance per second

(Depends on ω and radius)Same Rotational Speed

Different Tangential Speeds


Check Yourself

A pair of wheels are connected by a chain.Which wheel has higher rotational speed?

By how much?

Faster tangential speed on rim?1

2


Bicycles

FASTSLOW

Notice how gears work on a modern bicycle

Physics 101

Acceleration in Rotational MotionIf the angular velocity of a rotating object changes, it has an angular acceleration. Each point on the object has a corresponding tangentialacceleration.

Even if the angular velocity is constant, each point on the object has a centripetal acceleration because the velocity direction is changing.


Simple & Complex Objects

Motion of simple objects: PositionMotion of complex objects: Position & Rotation

SIMPLE COMPLEX

Axis ofRotation


Mass is a measure of inertia for linear motion.

Rotational inertia is similar concept for rotation.

Rotational Inertia

M mGold brick Normal brick

Difficult to start moving Easy to accelerate

x xWood Bat Plastic Pee-wee Bat

Difficult to Start Rotating Easy to Start Rotating

Physics 101

Rotational Inertia• An object rotating about an axis tends to

remain rotating about the same axis at the same rotational speed unless interfered with by some external influence.

• The property of an object to resist changes in its rotational state of motion is called rotational inertia.


Rotational Inertia

Rotational inertia depends on • Total mass of the object• Distribution of the mass relative to axis

Farther the mass is from the axis of rotation, the larger the rotational inertia.

Rotational inertia goes as (mass) x (axis_distance)2

Physics 101

Rotational InertiaDepends upon• mass of object.• distribution of mass

around axis of rotation.– The greater the distance

between an object’s mass concentration and the axis, the greater the rotational inertia.

Physics 101

Rotational Inertia• The greater the rotational inertia of an

object, the harder to change object’s rotational state.– A tightrope walker carries a long pole that has a high

rotational inertia, so it does not easily rotate.– Keeps the tightrope walker stable.

Physics 101

Rotational InertiaDepends upon the axis

around which it rotates• Easier to rotate pencil

around an axis passing through it.

• Harder to rotate it around vertical axis passing through center.

• Hardest to rotate it around vertical axis passing through the end.

Physics 101

Check Yourself

Which dancer has greater rotational inertia?

Axis of Rotation

A B


Demo: Long Legs

Long legs have greater rotational inertia than short legs so long legged animals have a slow walking stride.


Demo: Ramp Racing

The two disks and rods have the same mass but one has connecting rods near the center while the other has rods near the rim.

When rolled down a ramp, which wins the race?

Winner


Demo: Hoop & Disk Racing

Roll a hoop and a disk down a ramp; which wins the race?


When a force tends to cause a rotation, we say the force produces a torque.

Torque depends on• Magnitude of Force• Direction of Force• Lever Arm

(Torque) = (Force) x (Lever Arm)Positive torque gives counterclockwise rotation

Torque

Physics 10114-Sep-10 Physics 1 (Garcia) SJSU

Lever ArmLever arm is perpendicular distance from axis of rotation to the direction of the force.


Check Yourself

Identify the lever arm for each wrench.

ForceForce

Which wrench design gives more torque for the same force?

Physics 101

Torque for a Pirouette

The farther the distance between the feet, the greater the lever arm, so the greater the torque for creating the rotation.

Push onFloor

ReactionForce

LeverArm

Feet apart Feet together


Check Yourself

In which case are you exerting more torque?

A

BLever ArmLever Arm

almost zero


Balance of TorquesTorques clockwise and counter-clockwise

balance in both cases since(250 N) x (3 m) = (500 N) x (1.5 m)

Her lever armis still 3 meters


Center of Mass/Gravity

Average position of all the mass in an object is called the center of mass (CM) of object.

Average position of the weight distribution is called the center of gravity (CG).

When gravity is constant (usually the case), these two locations are the same.


Locating Center of Gravity

Balance an object to find center of gravity

Centerof

Gravity

Physics 101

Key Points of Lecture 9Key Points of Lecture 9

Before next lecture, read Hewitt through 1st half Chap.8

Homework Assignment #5 is due before 11:00 PM on Thursday, Sept. 16.

Homework Assignment #6 is due before 11:00 PM on Sunday, Sept. 19.

• Rotational Motion• Angular position• Angular (rotational) velocity• Angular acceleration• Rotational Inertia• Torque• Torque Balance• Center of Mass


Stability

Object is stable if CG is above the base.

CG CG

Wei

ght

Wei

ght

BASEAxis

AxisBASE

STABLE

UNSTABLE


Demo: Balance the CanIf a small amount of water is added to an

empty soda can, then the can may be balanced on its bottom edge.

x CG

PEPS

I


Check Yourself

Three trucksare parked ona slope. Whichtruck(s) tipover?

CG

CG

CG

BASEBASE

lecture 9 chapter 8 rotational motion - sfsu physics &...

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