physics 211: lecture 17 - ieu.edu.trhomes.ieu.edu.tr/hozcan/phys100/lect17.pdf · physics 211:...

Physics 211: Lecture 17, Pg 1

Physics 211: Lecture 17

Today’s Agenda

Rotational Kinematics

Analogy with one-dimensional kinematics

Kinetic energy of a rotating system

Moment of inertia

Discrete particles

Continuous solid objects

Parallel axis theorem


Rotation

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, Act 1 Rotations

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) the same as Bonnie’s

(b) twice Bonnie’s

(c) half Bonnie’s


Lecture 17, Act 1 Rotations

The angular velocity w of any point on a solid object rotating about a fixed axis is the same.

Both Bonnie & Klyde go around once (2p radians) every two seconds.

w

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

BonnieKlyde V2

1V


Rotational Variables.

Rotation about a fixed axis:

Consider a disk rotating about an axis through its center:

First, recall what we learned about Uniform Circular Motion:

(Analogous to )

w

dt

dw

dt

dxv

Spin round

blackboard


Rotational Variables...

Now suppose w can change as a function of time:

We define the angular acceleration:

w

2

2

dt

d

dt

d w

Consider the case when is constant.

We can integrate this to find w and as a function of time:

tww

0

constant

2

002

1tt w


Rotational Variables...

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

x = R

v = wR

And taking the derivative of this we find:

a = R

w

R

v

x 2

00

0

t2

1t

t

w

ww

constant


Summary (with comparison to 1-D kinematics)

Angular Linear

constant

t0 ww

w 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 = R v = wR a = R


Example: 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 = 2p radians)

a

R


Wheel And Rope...

a

R


Rotation & 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:

r1

r2 r3

r4

m4

m1

m2

m3

w


Rotation & Kinetic Energy...

So: but vi = wri

r1

r2 r3

r4

m4

m1

m2

m3

w v4

v1

v3

v2

K m vi ii

1

2

2

K m r m ri ii

i ii

1

2

1

2

2 2 2w w

which we write as:

K 1

2

2I w

I m ri ii

2

Define the moment of inertia

about the rotation axis I has units of kg m2.


Rotation & Kinetic Energy...

The kinetic energy of a rotating system looks similar to that of a point particle: Point Particle Rotating System

K 1

2

2I w

I m ri ii

2

K mv1

2

2

v is “linear” velocity

m is the mass.

w is angular velocity

I is the moment of inertia

about the rotation axis.


Moment 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 1

2

2I w I m ri ii

2

Inertia Rods

So where


Calculating 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:

m m

m m

L


Calculating Moment of Inertia...

m m

m m

L

r

L/2



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

m m

m m

L

r



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

m m

m m

L

r



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

L

I = 2mL2 I = mL2

m m

m m

I = 2mL2


Lecture 17, Act 2 Moment 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) Ia > Ib > Ic

(b) Ia > Ic > Ib

(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 an integral to find I :

I

m ri ii

N2

1

r

dm

I r dm2


Moments 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 1

2

2MR

Thin hoop of mass M and radius R,

about an axis through a diameter.

R

Hoop


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 2

5

2MR

R

I 1

2

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...

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

12

2ML

L

Thin rod of mass M and length L, about

a perpendicular axis through its end.

I 1

3

2ML

L

Rod


Parallel 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: 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/2 M

x CM

ICM IEND


Connection with CM motion

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

2CM

2iiNET MV

2

1um

2

1K

KREL KCM

KREL CM1

2

2I w

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

2

iiREL um2

1K Substituting ii ru w

2ii

2REL rm

2

1K w but CM

2ii rm I


Connection with CM motion...

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

2CM

2CMNET MV

2

1

2

1K wI

w

VCM

We will use this formula more in coming lectures.


Recap 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)

physics 211: lecture 17 - ieu.edu.trhomes.ieu.edu.tr/hozcan/phys100/lect17.pdf · physics 211:...

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