General Physics I, Lec 18, By/ T. A. Eleyan 1 Lecture 18 Rotational Motion.

General Physics I, Lec 18, By/ T. A. Eleyan

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Lecture 18Rotational Motion


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For circular motion, the distance (arc length) s, the radius r, and the angle are related by:

r

s

DEGRAD 180

Note that is measured in radians:

Angular Position, θ

> 0 for counterclockwise rotation from reference line


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The angular displacement is defined as the angle the object rotates through during some time interval

Every point on the disc undergoes the same angular displacement in any given time interval

if


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Notice that as the disk rotates, changes. We define the angular displacement, , as:

= f - i

which leads to the average angular speed av

if

ifav ttt

Angular Velocity, ω


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Instantaneous Angular Velocity

As usual, we can define the instantaneous angular velocity as:

tt

lim0

Note that the SI units of are: rad/s = s-1

> 0 for counterclockwise rotation < 0 for clockwise rotation

If v = speed of a an object traveling around a circle of radius r

= v / r


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The period of rotation is the time it takes to complete one revolution.

T

22

T

Problem:

What is the period of the Earth’s rotation about its own axis?

What is the angular velocity of the Earth’s rotation about its own axis?


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We can also define the average angular acceleration av:

And instantaneous angular acceleration

if

ifav ttt

tt

lim0

Angular Acceleration, a

The SI units of are: rad/s2 = s-2

We will skip any detailed discussion of angular acceleration, except to note that angular acceleration is the time rate of change of angular velocity


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Notes about angular kinematics:

When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration

i.e. , and are not dependent upon r, distance form hub or axis of rotation


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Examples:

1. Bicycle wheel turns 240 revolutions/min. What is its angular velocity in radians/second?

secradians1.25secradians8rev1

rads2

sec60

min1

min

rev240

2. If wheel slows down uniformly to rest in 5 seconds, what is the angular acceleration?

2secrad5sec5

secrad250

t

if


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3. How many revolution does it turn in those 5 sec?

srevolution102

rev1rad5.62)(

rad5.62sec5secrad52

1sec5secrad25

2

1

2

20

rev

tt

Recall that for linear motion we had:

Perhaps something similar for angular quantities?

20 2

1attvx


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Analogies Between Linear and Rotational Motion

Rotational Motion About a Fixed Axis with Constant Acceleration

Linear Motion with Constant Acceleration

ti

2

2

1tti

222i xavv i 222

2

2

1attvx i

atvv i


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Relationship Between Angular and Linear Quantities

Displacements

Speeds

Accelerations

r

s

vr

t

s

rt1

1

ra


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v r

Total linear acceleration is

Since the tangential speed v is

Tangential and the radial acceleration.

ta

The magnitude of tangential acceleration at is

The radial or centripetal acceleration ar is

ra

a

v

t

r

t

rt

r

r

v2

r

r 2

2r

22rt aa 222 rr 42 r


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Example: (a) What is the linear speed of a child seated 1.2m from the center of a steadily rotating merry-go-around that makes one complete revolution in 4.0s? (b) What is her total linear acceleration?

v

1 21.6 /

4.0 4.0

rev radrad s

s s

Since the angular speed is constant, there is no angular acceleration.

Tangential acceleration is ta

Radial acceleration is ra

a

r 1.2 1.6 / 1.9 /m rad s m s

r 2 21.2 0 / 0 /m rad s m s 2r 2 21.2 1.6 / 3.1 /m rad s m s

2 2t ra a 2 20 3.1 3.1 /m s


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Calculation of Moments of InertiaMoments of inertia for large objects can be computed, if we assume that the object consists of small volume elements with mass, mi.

It is sometimes easier to compute moments of inertia in terms of volume of the elements rather than their mass

Using the volume density, , replace dm in the above equation with dV.

The moment of inertia for the large rigid object is:

iii

mmrI

i

2

0lim dmr 2

dV

dm

The moments of inertia becomes

dVrI 2

dVdm


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The moment of inertia of a uniform hoop of mass M and radius R about an axis perpendicular to the plane of the hoop and passing through its center.

x

y

RO

dm

The moment of inertia is

dmrI 2

The moment of inertia for this object is the same as that of a point of mass M at the distance R.

dmR2 2MR


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Example for Rigid Body Moment of Inertia

Calculate the moment of inertia of a uniform rigid rod of length L and mass M about an axis perpendicular to the rod and passing through its center of mass.

The line density of the rod is L

M

so the mass is dm

I

dmrI 2

dx dxL

M

dmr 2 dxL

MxL

L2/

2/

2 2/

2/

3

3

1L

L

xL

M

33

223

LL

L

M

43

3L

L

M

12

2ML

dxL

MxL

0

2 L

xL

M

0

3

3

1

03

3 LL

M 3

3L

L

M

3

2ML


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Torque: The ability of a force to rotate a body about some axis .

Only the component of the force that is perpendicular to the radius causes a torque.

= r (Fsin)

Equivalently, only the perpendicular distance between the line of force and the axis of rotation, known as the moment arm r, can be used to calculate the torque.

= rF = (rsin)F


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The net torque about a point O is the sum of all torques about O:

221121 dFdF


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Problem:Calculate the net torque on the 0.6-m rod about the nail at the left. Three forces are acting on the rod as shown in the diagram.

30°

0.3 m

4 N

5 N

6 N


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Torque & Angular Acceleration

Let’s consider a point object with mass m rotating on a circle.

The tangential force Ft is tt maF

The torque due to tangential force Ft is

rFt I

Torque acting on a particle is proportional to the angular acceleration.

Analogs to Newton’s 2nd law of motion in rotation.

mr

rmat 2mr I


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How about a rigid object?

The external tangential force dFt is

tdF

The torque due to tangential force Ft is

The total torque is

d

tdma dmr

rdFt dmr 2

dmr 2 I


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Example:

A uniform rod of length L and mass M is attached at one end to a frictionless pivot and is free to rotate about the pivot in the vertical plane. The rod is released from rest in the horizontal position. What are the initial angular acceleration of the rod and the initial linear acceleration of its right end?


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The only force generating torque is the gravitational force Mg

Using the relationship between tangential and angular acceleration

L

dmrI0

2

Since the moment of inertia of the rod when it rotates about one end

We obtain

ta The tip of the rod falls faster than an object undergoing a free fall.

Fd2

LF

2

LMg I

L

dxx0

2L

x

L

M

0

3

3

3

2ML

I

MgL

2

32 2ML

MgL

L

g

2

3

L2

3g


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Rotational Kinetic EnergyKinetic energy of a rigid object that undergoing a circular motion:

Since a rigid body is a collection of mass, the total kinetic energy of the rigid object is

The moment of Inertia, I, is defined as

Kinetic energy of a mass mi, moving at a tangential speed, vi, is

iK

RK

i

iirmI 2

IKR 2

1The above expression is simplified to

2

2

1iivm 2

2

1iirm

i

iK i

iirm 22

1

iiirm 2

2

1


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Example:

In a system consists of four small spheres as shown in the figure, assuming the radii are negligible and the rods connecting the particles are massless, compute the moment of inertia and the rotational kinetic energy when the system rotates about the y-axis at .

x

y

M Ml l

m

m

b

bO


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I

Since the rotation is about y axis, the moment of inertia about y axis, Iy, is

RK

Thus, the rotational kinetic energy is

Find the moment of inertia and rotational kinetic energy when the system rotates on the x-y plane about the z-axis that goes through the origin O.

Why are some 0s?

This is because the rotation is done about y axis, and the radii of the spheres are negligible.

2i

iirm 2Ml 22Ml

2

2

1 I 2222

1 Ml 22Ml

2Ml 20m 20m

General Physics I, Lec 18, By/ T. A. Eleyan 1 Lecture 18 Rotational Motion.

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