1

Chapter 8

Rotational Motion8-1 Angular Quantities

•In purely rotational motion, allpoints on the object move in circlesaround the axis of rotation (“O”).

•All points on a straight line drawnthrough the axis move through thesame angle in the same time.

•The angle θ in radians is defined:

where l is the arc length.

(8-1a)

Think-Pair-Share

• Problem 1: Express the following angle inradians: (a) 30 degrees, (b) 57 degrees.Give as numerical values and as fractionsof pi.

8-1 Angular Quantities•Angular displacement:

•Average angular velocity:

•The instantaneous angularvelocity:

(8-2a)

(8-2b)

2

8-1 Angular Quantities

The angular acceleration is the rate at which theangular velocity changes with time:

The instantaneous acceleration:

(8-3a)

(8-3b)

Think-Pair-Share

• Problem 4: The blades of a blender rotateat a rate of 6500 rpm. When the motor isturned off during operation, the bladesslow to rest in 3.0 s. What is the angularacceleration as the blades slow down?

8-1 Angular QuantitiesEvery point on a rotating body has an angularvelocity ω and a linear velocity v.

They are related: (8-4)

8-1 Angular Quantities

Therefore, objectsfarther from the axisof rotation will movefaster.

3

8-1 Angular Quantities

If the angular velocity of arotating object changes, ithas a tangentialacceleration:

Even if the angularvelocity is constant, eachpoint on the object has acentripetal acceleration:

(8-5)

(8-6)

8-1 Angular Quantities

Here is the correspondence between linearand rotational quantities:

8-1 Angular Quantities

The frequency is the number of completerevolutions per second:

Frequencies are measured in hertz.

The period is the time one revolution takes:

(8-8)

Think-Pair-Share

• Problem 5: A child rolls a ball on a levelfloor 3.5 m to another child. If the ballmakes 15.0 revolutions, what is itsdiameter?

4

8-2 Constant Angular Acceleration

The equations of motion for constant angularacceleration are the same as those for linearmotion, with the substitution of the angularquantities for the linear ones.

Think-Pair-Share

• Problem 15: A centrifuge acceleratesuniformly from rest to 15,000 rpm in 220 s.Through how many revolutions did it turnin this time?

8-3 Rolling Motion (Without Slipping)• Rolling without slipping - static friction

•wheel contact point is a rest WRT ground

•The linear speed of the wheel is related to itsangular speed:

8-4 TorqueTo make an object start rotating, a force is needed;the position and direction of the force matter as well.

The perpendicular distance from the axis of rotationto the line along which the force acts is called thelever arm.

5

8-4 Torque

A longer leverarm is veryhelpful inrotating objects.

8-4 Torque

•Torque = rotational analog of force

•r = distance from point of rotation to pointwhere force is applied

•F = force perpendicular to r

•FA applies maximum torque

Think-Pair-Share

• Problem 22: A 55-kg person riding a bikeputs all her weight on each pedal whenclimbing a hill. The pedals rotate in acircle of radius 17 cm. (a) What is themaximum torque she exerts?

8-5 Rotational Dynamics; Torque andRotational Inertia

Knowing that , we see that

This is for a single pointmass; what about anextended object?

As the angularacceleration is the samefor the whole object, wecan write:

(8-11)

(8-12)

6

8-5 Rotational Dynamics; Torque andRotational Inertia

The quantity is called therotational inertia of an object.

The distribution of mass matters here – thesetwo objects have the same mass, but the one onthe left has a greater rotational inertia, as somuch of its mass is far from the axis of rotation.

8-5 RotationalDynamics; Torque

and RotationalInertia

The rotational inertia ofan object depends notonly on its massdistribution but also thelocation of the axis ofrotation – compare (f)and (g), for example.

8-5 Rotational Inertia

• I = moment of inertia– rotational equivalent of mass– changes with axis of rotation

• Rotational equivalent of Newton’s 2nd Law:

Think-Pair-Share

• Problem 27: Determine the moment ofinertia of a 10.8-kg sphere of radius 0.648m when the axis of rotation is through itscenter.

7

8-6 Solving Problems in RotationalDynamics

1. Draw a diagram.

2. Decide what the system comprises.

3. Draw a free-body diagram for each objectunder consideration, including all the forcesacting on it and where they act.

4. Find the axis of rotation; calculate the torquesaround it.

5. Apply Newton’s second law for rotation. Ifthe rotational inertia is not provided, youneed to find it before proceeding with thisstep.

6. Apply Newton’s second law for translationand other laws and principles as needed.

7. Solve.

8. Check your answer for units and correctorder of magnitude.

8-6 Solving Problems in RotationalDynamics