ROTASI

Physics 111: Lecture 17, Pg 1

Physics 111: Lecture 17Physics 111: Lecture 17

Today’s AgendaToday’s Agenda Rotational Kinematics

Analogy with one-dimensional kinematics

Kinetic energy of a rotating system Moment of inertiaDiscrete particlesContinuous solid objects

Parallel axis theorem


RotationRotation

Up until now we have gracefully avoided dealing with the rotation of objects. We have studied objects that slide, not roll.We have assumed pulleys are without mass.

Rotation is extremely important, however, and we need to understand it!

Most of the equations we will develop are simply rotational analogues of ones we have already learned when studying linear kinematics and dynamics.


Lecture 17, Lecture 17, Act 1Act 1RotationsRotations

Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds.Klyde’s angular velocity is:

(a)(a) the same as Bonnie’s (b)(b) twice Bonnie’s

(c)(c) half Bonnie’s


Lecture 17, Lecture 17, Act 1Act 1RotationsRotations

The angular velocity of any point on a solid object rotating about a fixed axis is the same.Both Bonnie & Klyde go around once (2 radians) every two seconds.

(Their “linear” speed v will be different since v = r).

BonnieKlyde V21V


Rotational Variables.Rotational Variables.

Rotation about a fixed axis:Consider a disk rotating about

an axis through its center:

First, recall what we learned aboutUniform Circular Motion:

(Analogous to )

dtd

dtdxv

Spin round

blackboard


Rotational Variables...Rotational Variables...

Now suppose can change as a function of time: We define the

angular acceleration:

2

2

dtd

dtd

Consider the case when is constant. We can integrate this to find and as a function of time:

t

0

constant

200 2

1 tt


Rotational Variables...Rotational Variables...

Recall also that for a point at a distance R away from the axis of rotation:x = Rv = R

And taking the derivative of this we find:a = R

R

v

x 2

00

0

t21t

t

constant


Summary Summary (with comparison to 1-D kinematics)(with comparison to 1-D kinematics)

Angular Linear

constant

t0

0 021

2t t

constanta

v v at 0

x x v t at 0 021

2

And for a point at a distance R from the rotation axis:

x = Rv = Ra = R


Example: Wheel And RopeExample: Wheel And Rope A wheel with radius R = 0.4 m rotates freely about a fixed axle. There is a rope wound around the wheel. Starting from rest at t = 0, the rope is pulled such

that it has a constant acceleration a = 4 m/s2. How many revolutions has the wheel made after 10 seconds? (One revolution = 2 radians)

aa

R


Wheel And Rope...Wheel And Rope... Use a = R to find :

= a / R = 4 m/s2 / 0.4 m = 10 rad/s2

Now use the equations we derived above just as you would use the kinematic equations from the beginning of the semester.

rev80radrot

21x rad500

20 t

21t = 0 + 0(10) + (10)(10)2 = 500 rad2

10

aa

R


Rotation & Kinetic EnergyRotation & Kinetic Energy

Consider the simple rotating system shown below. (Assume the masses are attached to the rotation axis by massless rigid rods).

The kinetic energy of this system will be the sum of the kinetic energy of each piece:

rr1

rr2rr3

rr4

m4

m1

m2

m3


Rotation & Kinetic Energy...Rotation & Kinetic Energy...

So: but vi = ri

rr1

rr2rr3

rr4

m4

m1

m2

m3

vv4

vv1

vv3

vv2

K m vi ii

12

2

K m r m ri ii

i ii

12

12

2 2 2

which we write as:

K 12

2I

I m ri ii

2

Define the moment of inertiamoment of inertiaabout the rotation axis I has units of kg m2.


Rotation & Kinetic Energy...Rotation & Kinetic Energy...

The kinetic energy of a rotating system looks similar to that of a point particle:

Point ParticlePoint Particle Rotating System Rotating System

K 12

2I

I m ri ii

2

K mv12

2

v is “linear” velocitym is the mass.

is angular velocityI is the moment of inertiaabout the rotation axis.


Moment of InertiaMoment of Inertia

Notice that the moment of inertia I depends on the distribution of mass in the system.The further the mass is from the rotation axis, the bigger the moment of inertia.

For a given object, the moment of inertia will depend on where we choose the rotation axis (unlike the center of mass).

We will see that in rotational dynamics, the moment of inertia I appears in the same way that mass m does when we study linear dynamics!

K 12

2I I m ri ii

2Inertia Rods

So where


Calculating Moment of InertiaCalculating Moment of Inertia

We have shown that for N discrete point masses distributed about a fixed axis, the moment of inertia is:

I m ri ii

N2

1

where r is the distance from the mass to the axis of rotation.

Example: Calculate the moment of inertia of four point masses(m) on the corners of a square whose sides have length L, about a perpendicular axis through the center of the square:

mm

mm

L


Calculating Moment of Inertia...Calculating Moment of Inertia...

The squared distance from each point mass to the axis is:

mm

mm

Lr

L/2

2L

2L2r

222

2Lm4

2Lm

2Lm

2Lm

2LmrmI

22222N

1i

2ii

so

I = 2mL2

Using the Pythagorean Theorem



Now calculate I for the same object about an axis through the center, parallel to the plane (as shown):

mm

mm

L

r

4Lm4

4Lm

4Lm

4Lm

4LmrmI

22222N

1i

2ii

I = mL2



Finally, calculate I for the same object about an axis along one side (as shown):

mm

mm

L

r

2222N

1i

2ii 0m0mmLmLrmI

I = 2mL2



For a single object, I clearly depends on the rotation axis!!

L

I = 2mL2I = mL2

mm

mm

I = 2mL2


Lecture 17, Lecture 17, Act 2Act 2Moment of InertiaMoment of Inertia

A triangular shape is made from identical balls and identical rigid, massless rods as shown. The moment of inertia about the a, b, and c axes is Ia, Ib, and Ic respectively.Which of the following is correct:

(a)(a) Ia > Ib > Ic

(b)(b) Ia > Ic > Ib

(c)(c) Ib > Ia > Ic

a

b

c



a

b

c

Label masses and lengths:

m

m m

L

L

Ia m L m L mL 2 2 82 2 2

Calculate moments of inerta:

Ib mL mL mL mL 2 2 2 23

Ic m L mL 2 42 2

So (b) is correct: Ia > Ic > Ib



For a discrete collection of point masses we found:

For a continuous solid object we have to add up the mr2 contribution for every infinitesimal mass element dm.

We have to do anintegral to find I :

I m ri ii

N2

1

r

dm

I r dm2


Moments of InertiaMoments of Inertia

Some examples of I for solid objects:

Thin hoop (or cylinder) of mass M and radius R, about an axis through its center, perpendicular to the plane of the hoop.

I MR 2

R

I 12

2MRThin hoop of mass M and radius R, about an axis through a diameter.

R

Hoop


Moments of Inertia...Moments of Inertia...

Some examples of I for solid objects:

Solid sphere of mass M and radius R, about an axis through its center.

I 25

2MR

R

I 12

2MR

R

Solid disk or cylinder of mass M and radius R, about a perpendicular axis through its center.

Sphere and disk



Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold.Which one has the biggest moment of inertia about an axis through its center?

same mass & radius

solid hollow

(a) solid aluminum(b) hollow gold (c) same



Moment of inertia depends on mass (same for both) and distance from axis squared, which is bigger for the shell since its mass is located farther from the center.The spherical shell (gold) will have a bigger moment of inertia.

same mass & radius

ISOLID < ISHELL

solid hollow


Moments of Inertia...Moments of Inertia...

Some examples of I for solid objects (see also Tipler, Table 9-1):

Thin rod of mass M and length L, about a perpendicular axis through its center.

I 1

122ML

L

Thin rod of mass M and length L, about a perpendicular axis through its end.

I 13

2ML

L

Rod


Parallel Axis TheoremParallel Axis Theorem

Suppose the moment of inertia of a solid object of mass M about an axis through the center of mass, ICM, is known. The moment of inertia about an axis parallel to this axis but a distance D away is given by:

IPARALLEL = ICM + MD2

So if we know ICM , it is easy to calculate the moment of inertia about a parallel axis.


Parallel Axis Theorem: ExampleParallel Axis Theorem: Example

Consider a thin uniform rod of mass M and length D. Figure out the moment of inertia about an axis through the end of the rod.

IPARALLEL = ICM + MD2

L

D=L/2M

xCM

ICM ML1

122We know

IEND ML M L ML

112 2

13

22

2So

which agrees with the result on a previous slide.

ICMIEND


Connection with CM motionConnection with CM motion

Recall what we found out about the kinetic energy of a system of particles in Lecture 15:

2CM

2iiNET MV

21um

21K

KREL KCM

KREL CM12

2I

For a solid object rotating about its center of mass, we now see that the first term becomes:

2iiREL um

21K Substituting ii ru

2ii

2REL rm

21K but CM

2ii rm I


Connection with CM motion...Connection with CM motion...

So for a solid object which rotates about its center or mass and whose CM is moving:

2CM

2CMNET MV

21

21K I

VCM

We will use this formula more in coming lectures.


Recap of today’s lectureRecap of today’s lecture

Rotational Kinematics (Text: 9-1) Analogy with one-dimensional kinematics

Kinetic energy of a rotating system Moment of inertia (Text: 9-2, 9-3, Table

9-1)Discrete particles (Text: 9-3)Continuous solid objects (Text: 9-3)

Parallel axis theorem (Text: 9-3)

Look at textbook problems Look at textbook problems Chapter 9: # 7, 11, 27, 31, 33, 37

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