Ch. 8: Rotational Equilibrium and Dynamics

Objectives

• Know the distinction between translational motion and rotational motion.

• Understand the concept of torque and be able to make related calculations.

Rolling Race

Roll various objects down a ramp: spheres, solid cylinders, hollow cylinders, and washers. Compare which objects are the fastest. Develop a hypothesis regarding what factor(s) affect the relative speeds of the different objects.

Masses and Motion

point masses (center of mass) can havetranslational motion

extended masses can haverotational motion

Each type of motion can be analyzed separately.

Torque

• torque: the ability of a force to rotate an object around an axis (t)

• t = F·d·sinq• vector quantity• clockwise (─)• counterclockwise (+)• St = t1 + t2 + t3 + …

F

q

d

Net Torque Problem

Jack (244 N) and Bill (215 N) are sitting at opposite ends of a horizontal teeter-totter. If Jack is sitting 1.75 m from the center and Jill is sitting 1.95 m from the center, what is the net torque? What is the net torque if the teeter-totter is oriented upward at a 15o angle toward Jill’s end?

Objectives

• Understand the concept of “center-of-mass.”• Be able to find the center of mass for an

irregularly-shaped object.• Understand the concept of “moment of inertia.”• Be able to compare the moment of inertia for

differently-shaped objects.• Understand the concept of rotational

equilibrium and make related calculations.

Center of Mass

center of mass: the point around which an object rotates if gravity is only force acting (see video)

Center of Mass

“Fosbury Flop”

An object will “topple” once its centerof mass is no longer supported by a pivot.

Finding the Center of Mass

• Follow the directions for the “Quick Lab” on page 284.

• Predict the location of the center of mass before you proceed.

• You don’t need to write anything—just for fun.

Moment of Inertia

• moment of inertia (I): the tendency of an object to resist changes in rotational motion

• related to mass distribution

• This is why hoops accelerate slowly and spheres quickly

• torque needed to rotate differs (try book)

Moment of Inertia

Moment of Inertia Problem

• What is the moment of inertia of a 35 gram metal cylinder with r = 0.015 cm rolling down an incline?

Moment of Inertia Questions

• Does a single object have a single moment of inertia? Explain.

• What shape/axis would have the largest moment of inertia theoretically?

• Why do bicycles have such large, yet thin tires?

Rotational Equilibrium

• Translational equilibrium: SF = 0(no linear acceleration)

• Rotational equilibrium: St = 0(no rotational acceleration)

• Any axis can be used—choose for simplicity!

Rotational Equilibrium

A 5.55 N meter stick is suspended from two spring scales (one at each end). A 9.05 N mass is hung at the 65.0-cm mark. How much force is applied by each spring scale (scale A, scale B)?

Objectives

• Understand the concepts of angular speed and angular acceleration.

• Be able to make angular speed and angular acceleration calculations.

Radians

• Angles can be measured in “radians.”• • 1 radian = 57.3o

• 2 p rad = 360o

qr

s

Angular Speed

• speed = distance / time• angular speed = angular distance / time• or• measured in rad/s• What is the angular speed of a carousel with a period

of 8.5 seconds?

Angular to Tangential Speed

• tangential speed: • What is the tangential speed a child sitting 3.5 m

from the center of the carousel in the previous problem?

Angular Acceleration

• Angular acceleration is analogous to linear acceleration. It is a change in the rate of rotation.

• Tangential acceleration: • The angular speed of a camshaft increases from 145

rad/s to 528 rad/s in 0.75 s. What is a? What is tangential acceleration of the shaft (r = 0.052 m) at the end?

Objectives

• Understand and use Newton’s second law for rotation.

• Understand and apply the concept of angular momentum.

• Understand and apply the concept of rotational KE.

Second Law for Rotation

• Translational 2nd Law: • Rotational 2nd Law: • What is the angular acceleration of a 0.35 kg solid

sphere with radius 0.27 m if a 4.2 N net force is applied tangential to the surface?

Angular Momentum

• Translational momentum: • Rotational (angular momentum): • Conservation of Angular Momentum: • Why do skaters spin faster when they pull their arms

inward? Demo!• Remember electron spin? Electrons really don’t spin,

but they have quantized angular momentum.

Conservation Problem

A 0.11 kg mouse rides the edge of a Lazy Susan that has a mass of 1.3 kg and a radius of 0.25 m. If the angular speed is initially 3.0 rad/s, what is the angular speed after the mouse moves to a point 0.15 m from the center?

Rotational Kinetic Energy

• Translational KE: • Rotational KE:

Rotational KE Problems

A 1.5 kg solid sphere with radius 12 cm begins rolling down an incline. What is the translational speed of the sphere after it has dropped a vertical distance of 2.4 meters?

Objectives

• Be able to identify simple machines.• Be able to explain how simple machines make

doing work “easier.”• Be able to calculate the ideal mechanical

advantage (IMA), actual mechanical (AMA) advantage, input work (WA), output work (WO), and efficiency (e) of a simple machine.

Simple Machines

Simple machines generally make doing work easier by reducing applied force (but distance is increased).

4 kinds: lever, inclined plane, pulley, wheel and axle

input work: WA = FA·dA output work:

WO = FO·dO

If no friction: WA = WO

If friction is present: WA > WO

Simple Machines

O

A

ddIMA

A

O

FFAMA

mechanical advantage (MA): factor by whichinput force is multiplied by the machine

efficiency: ratio of output work to input work (indicates amount of friction in machine)

100WWe

A

O

“ideal” “actual”