Chapter 8 Rotational Motion. Objectives Distinguish between inertia and moment of inertia. Calculate the moment of inertia of various objects. Explain

Chapter 8

Rotational Motion

Objectives

Distinguish between inertia and moment of inertia.

Calculate the moment of inertia of various objects. Explain the meaning of the radius of gyration. Use the radius of gyration to solve for an object's moment of inertia.

Engagement

Discussion of Launch Activity

1. List objects from greatest to least acceleration.

2. Which of the object’s properties may have contributed to their behavior?

3. List properties that were the same and different for each object.

4. Demonstration

5. Demonstration 2

https://www.youtube.com/watch?v=cB8GNQuyMPc

https://www.youtube.com/watch?v=uyU25DdONjo

Which will roll down an incline with greater acceleration, a hollow cylinder or a solid cylinder of the same mass and radius? The answer is the cylinder with the smaller rotational inertia because the cylinder with the greater rotational inertia requires more time to get rolling.

Rotational Inertia and Rolling

Inertia of any kind is a measure of “laziness.” The cylinder with its mass concentrated farthest from the axis of rotation—the hollow cylinder—has the greater rotational inertia. The solid cylinder will roll with greater acceleration.


Any solid cylinder will roll down an incline with more acceleration than any hollow cylinder, regardless of mass or radius. A hollow cylinder has more “laziness per mass” than a solid cylinder.


A solid cylinder rolls down an incline faster than a hollow one, whether or not they have the same mass or diameter.


think!A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down an incline. Which will have more acceleration?


think!A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down an incline. Which will have more acceleration?

Answer:

The cylinders have different masses, but the same rotational inertia per mass, so both will accelerate equally down the incline. Their different masses make no difference, just as the acceleration of free fall is not affected by different masses. All objects of the same shape have the same “laziness per mass” ratio.


think!Would you expect the rotational inertia of a hollow sphere about its center to be greater or less than the rotational inertia of a solid sphere? Defend your answer.


think!Would you expect the rotational inertia of a hollow sphere about its center to be greater or less than the rotational inertia of a solid sphere? Defend your answer.

Answer:

Greater. Just as the value for a hoop’s rotational inertia is greater than a solid cylinder’s, the rotational inertia of a hollow sphere would be greater than that of a same-mass solid sphere for the same reason: the mass of the hollow sphere is farther from the center.


The shape of an object determines how easy or hard it is to spin

For objects of the same mass, the longerone is tougher to spin takes more torque

Hinge

It matters where the hinge is

The stick with the hinge at the end takes 4 timesmore torque to get it spinning than the stick withthe hinge in the center.

Rotational Inertia (moment of inertia)

• Rotational inertia is a parameter that is used to quantify how much torque it takes to get a particular object rotating

• it depends not only on the mass of the object, but where the mass is relative to the hinge or axis of rotation

• the rotational inertia is bigger, if more mass is located farther from the axis.

How fast does it spin?

• For spinning or rotational motion, the rotational inertia of an object plays the same role as ordinary mass for simple motion

• For a given amount of torque applied to an object, its rotational inertia determines its rotational acceleration the smaller the rotational inertia, the bigger the rotational acceleration

Newton’s first law, the law of inertia, applies to rotating objects.

• An object rotating about an internal axis tends to keep rotating about that axis.

• Rotating objects tend to keep rotating, while non-rotating objects tend to remain non-rotating.

• The resistance of an object to changes in its rotational motion is called rotational inertia (sometimes moment of inertia).

Rotational Inertia

Just as it takes a force to change the linear state of motion of an object, a torque is required to change the rotational state of motion of an object. In the absence of a net torque, a rotating object keeps rotating, while a non-rotating object stays non-rotating.

Rotational Inertia

Big rotationalinertia

Small rotationalinertia

Same torque,different

rotational inertia

spinsslow

spinsfast

Rotational Inertia and MassLike inertia in the linear sense, rotational inertia depends on mass, but unlike inertia, rotational inertia depends on the distribution of the mass. The greater the distance between an object’s mass concentration and the axis of rotation, the greater the rotational inertia.

Rotational Inertia

8-5 Rotational Dynamics; Torque and Rotational Inertia

Knowing that , we see that

This is for a single point mass; what about an extended object?

As the angular acceleration is the same for the whole object, we can write:

(8-11)

(8-12)

8-5 Rotational Dynamics; Torque and Rotational Inertia

The quantity is called the rotational inertia of an object.

The distribution of mass matters here – these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation.

Demo with inertial rods

Rotational inertia depends on the distance of mass from the axis of rotation.

Rotational Inertia

By holding a long pole, the tightrope walker increases his rotational inertia.

Rotational Inertia

A long baseball bat held near its thinner end has more rotational inertia than a short bat of the same mass.

• Once moving, it has a greater tendency to keep moving, but it is harder to bring it up to speed.

• Baseball players sometimes “choke up” on a bat to reduce its rotational inertia, which makes it easier to bring up to speed.

A bat held at its end, or a long bat, doesn’t swing as readily.

Rotational Inertia

The short pendulum will swing back and forth more frequently than the long pendulum.

Rotational Inertia

For similar mass distributions, short legs have less rotational inertia than long legs.

Rotational Inertia

The rotational inertia of an object is not necessarily a fixed quantity. It is greater when the mass within the object is extended from the axis of rotation.

Rotational Inertia

You bend your legs when you run to reduce their rotational inertia. Bent legs are easier to swing back and forth.

Rotational Inertia

think!When swinging your leg from your hip, why is the rotational inertia of the leg less when it is bent?

Rotational Inertia

think!When swinging your leg from your hip, why is the rotational inertia of the leg less when it is bent?

Answer:

The rotational inertia of any object is less when its mass is concentrated closer to the axis of rotation. Can you see that a bent leg satisfies this requirement?

Rotational Inertia

8-5 Rotational Dynamics; Torque

and Rotational Inertia

The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation – compare (f) and (g), for example.

Formulas for Rotational Inertia

When all the mass m of an object is concentrated at the same distance r from a rotational axis, then the rotational inertia is I = mr2. When the mass is more spread out, the rotational inertia is less and the formula is different.

Rotational Inertia

The human body can rotate freely about three principal axes of rotation.Each of these axes is at right angles to the others and passes through the center of gravity. The rotational inertia of the body differs about each axis.

Rotational Inertia and Gymnastics

The human body has three principal axes of rotation.


Longitudinal Axis

Rotational inertia is least about the longitudinal axis, which is the vertical head-to-toe axis, because most of the mass is concentrated along this axis.

• A rotation of your body about your longitudinal axis is the easiest rotation to perform.

• Rotational inertia is increased by simply extending a leg or the arms.

• Period 1 stopped here.


An ice skater rotates around her longitudinal axis when going into a spin.

a.The skater has the least amount of rotational inertia when her arms are tucked in.


An ice skater rotates around her longitudinal axis when going into a spin.

a.The skater has the least amount of rotational inertia when her arms are tucked in.

b.The rotational inertia when both arms are extended is about three times more than in the tucked position.


c and d. With your leg and arms extended, you can vary your spin rate by as much as six times.


Transverse Axis

You rotate about your transverse axis when you perform a somersault or a flip.


A flip involves rotation about the transverse axis.a. Rotational inertia is least in the tuck position.

12.2 Rotational Inertia and Gymnastics

A flip involves rotation about the transverse axis.a. Rotational inertia is least in the tuck position. b. Rotational inertia is 1.5 times greater.

12.2 Rotational Inertia and Gymnastics

A flip involves rotation about the transverse axis.a. Rotational inertia is least in the tuck position. b. Rotational inertia is 1.5 times greater. c. Rotational inertia is 3 times greater.


A flip involves rotation about the transverse axis.a. Rotational inertia is least in the tuck position. b. Rotational inertia is 1.5 times greater. c. Rotational inertia is 3 times greater. d. Rotational inertia is 5 times greater than in the tuck position.


Rotational inertia is greater when the axis is through the hands, such as when doing a somersault on the floor or swinging from a horizontal bar with your body fully extended.


The rotational inertia of a body is with respect to the rotational axis.

a.The gymnast has the greatest rotational inertia when she pivots about the bar.


The rotational inertia of a body is with respect to the rotational axis.

a.The gymnast has the greatest rotational inertia when she pivots about the bar.

b.The axis of rotation changes from the bar to a line through her center of gravity when she somersaults in the tuck position.


The rotational inertia of a gymnast is up to 20 times greater when she is swinging in a fully extended position from a horizontal bar than after dismount when she somersaults in the tuck position. Rotation transfers from one axis to another, from the bar to a line through her center of gravity, and she automatically increases her rate of rotation by up to 20 times. This is how she is able to complete two or three somersaults before contact with the ground.


Medial Axis

The third axis of rotation for the human body is the front-to-back axis, or medial axis. This is a less common axis of rotation and is used in executing a cartwheel.


8-6 Solving Problems in Rotational Dynamics

1. Draw a diagram.

2. Decide what the system comprises.

3. Draw a free-body diagram for each object under consideration, including all the forces acting on it and where they act.

4. Find the axis of rotation; calculate the torques around it.

5. Apply Newton’s second law for rotation. If the rotational inertia is not provided, you need to find it before proceeding with this step.

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

7. Solve.

8. Check your answer for units and correct order of magnitude.

8-6 Solving Problems in Rotational Dynamics

Elaboration

Moments of Inertia hand-out

Practice Problem 1 (#28 in text)

Calculate the moment of inertia of a 66.7cm-diameter bicycle wheel. The rim and tire have a combined mass of 1.25kg. The mass of the hub can be ignored (why?).

Practice Problem 1 (#28 in text)Calculate the moment of inertia of a 66.7cm-diameter bicycle wheel. The rim and tire have a combined mass of 1.25kg. The mass of the hub can be ignored (why?).

The moments of inertia are listed on p. 223, and a thin hoop through its center is:I = mr2

m = 1.25 kgr = (.667 m)/2 = .3335 m (They gave you the diameter)so I = (1.25 kg)(.3335 m)2 = 0.139 kgm2

Practice Problem 2 (similar to #29)

A small 1.05kg ball on the end of the light rod is rotated in a horizontal circle of a radius 0.900 m. Calculate (a) the moment of inertia of the system about the axis of rotation, and (b) the torque needed to keep the ball rotating at a constant angular velocity if air resistance exerts a force of 0.0800N on the ball.

Practice Problem 2 (similar to #29)A small 1.05kg ball on the end of the light rod is rotated in a horizontal circle of a radius 0.900 m. Calculate (a) the moment of inertia of the system about the axis of rotation, and (b) the torque needed to keep the ball rotating at a constant angular velocity if air resistance exerts a force of 0.0800N on the ball.

The small ball can be treated as a particle for calculating the moment of inertia.

I = mr2

I = (1.05 kg)(.900)2 = 0.8505 kgm2

If the mass is moving at a constant angular velocity, then it is not accelerating (net torque is zero), and the only necessary torque needed is the same magnitude as the torque caused by the frictional force of .0800 N acting at a radius of .900 m on the ball:Since force is applied at a 90o angle to the radius, so the factor sinbecomes 1, and really the torque is:(.0800 N)(.900 m) = .072 Nm

Practice Problem 3 (#40)A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8-42. If each of the three rotor helicopter blades is 3.75m long and has a mass of 160kg, calculate the moment of inertia of the three rotor blades about the axis of rotation. How much torque must the motor apply to bring the blades up to a speed of 5.0rev/s in 8.0s?

Practice Problem 3 (#40)A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8-42. If each of the three rotor helicopter blades is 3.75m long and has a mass of 160kg, calculate the moment of inertia of the three rotor blades about the axis of rotation. How much torque must the motor apply to bring the blades up to a speed of 5.0rev/s in 8.0s?

The moments of inertia are listed on p. 208, and a long thin rod through its end is:I = 1/3ML2

so each rotor has a moment of inertia of:I = 1/3(160 kg)(3.75 m)2 = 2500 kgm2 = 750 kgm2

Three rotors would have three times this moment:I = (750 kgm2)3 = 2250 kgm2

Now we need to solve a kinematics question:o = 0 = (5 Revolutions/s)(2radians/revolution) = 31.416 rad/st = 8.0 s

Now apply:o + t = 3.927 rad/s/s

And now find the torque using F = ma:= (2250 kgm2)(3.927 rad/s/s) = 8836 Nm = 8800 Nm of torque

http://tuhsphysics.ttsd.k12.or.us/Tutorial/NewIBPS/PS8_9/PS8_9.htm%23toc

Homework

• Chapter 8 Problems #27, 31, 33

Closure

• Kahoot 8-5 and 8-6

Documents

Chapter 8 Rotational Motion. Objectives Distinguish between inertia and moment of inertia. Calculate the moment of inertia of various objects. Explain