# Rotational motion Angular displacement, angular velocity, angular acceleration

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### Transcript of Rotational motion Angular displacement, angular velocity, angular acceleration

• Rotational motion• Angular displacement, angular velocity, angular acceleration• Rotational energy• Moment of Inertia• Torque

Chapter 10:Rotation of a rigid object about a fixed axis

Reading assignment: Chapter 10.1 to10.4, 10.5 (know concept of moment of inertia, don’t worry about integral calculation), 10.6 to 10.9

Homework 10.1 (due Tuesday, Oct. 23): CQ1, CQ2, AE1, 2, 3, 6, 7, 12

Homework 10.2 (due Wednesday, Oct. 24): CQ8, QQ3, QQ4, OQ3, OQ4, OQ6, OQ8, AE3, 13, 15, 19, 26, 29

Homework 10.3 (due Friday, Oct. 26): CQ13, 35, 36, 38, 43, 49, 55, 56, 59

Planar, rigid object rotating about origin O.

Rotational motion

Look at one point P:

rsArc length s:

Thus the angle (angular position) is: s

r

is measured in degrees or radians (SI unit: radian)

Full circle has an angle of 2p radians.

Thus, one radian is 360°/2p 57.3

Radian degrees2p 360°p 180°p/2 90°

1 57.3°

Define quantities for circular motion

(note analogies to linear motion!!)

Angular displacement:

Average angular speed:

Instantaneous angular speed:

Average angular acceleration:

Instantaneous angular acceleration:

if

ttt if

if

dtd

tt

0

lim

ttt if

if

dtd

tt

0

lim

Angular velocity is a vector

Right-hand rule for determining the direction of this vector.

• rotates through the same angle, • has the same angular velocity,• has the same angular acceleration.

Every particle (of a rigid object):

, , characterize rotational motion of entire object

Linear motion with constant linear acceleration, a.

tavv xxixf

2

21 tatvxx xxiif

)(222ifxxixf xxavv

tvvxx xfxiif )(21

Rotational motion with constant rotational acceleration, .

tif

2

21 ttiif

)(222ifif

tfiif )(21

Exactly the same equations, just different symbols!!

Black board example 10.1

A wheel starts from rest and rotates with constant angular acceleration and reaches an angular speed of 12.0 rad/s in 3.00 s.

1. What is the magnitude of the angular acceleration of the wheel (in rad/s2)?

A. 0

B. 1

C. 2

D. 3

E. 4

2. Through what angle does the wheel rotate in these 3 sec (in rad)?

A. 18

B. 24

C. 30

D. 36

E. 48

3. Through what angle does the wheel rotate between 2 and 3 sec (in rad)?

A. 5

B. 10

C. 15

D. 20

E. 25

Relation between angular and linear quantities

tv r Tangential speed of a point P:

ta r Tangential acceleration of a point P:

Note: This is not the centripetal acceleration ar

This is the tangential acceleration at

rsArc length s:

A fly is sitting at the end of a ceiling fan blade. The length of the blade is 0.50 m and it spins with 40.0 rev/min.

a) Calculate the (tangential) speed of the fly.

b) What are the tangential and angular speeds of another fly sitting half way in?

c) Starting from rest it takes the motor 20 seconds to reach this speed. What is the angular acceleration?

d) At the final speed, with what force does the fly (m = 0.01 kg, r = 0.50 m) need to hold on, so that it won’t fall off?

(Note difference between angular and centripetal acceleration).

Black board example 10.2

vt

Demo:

Both sticks have the same weight.

Why is it so much more difficult to rotate the blue stick?

Rotational kinetic energy

A rotating object (collection of i points with mass mi) has a rotational kinetic energy of

2

21 IKR

Where:2

ii

i rmI Moment of inertia or rotational inertia

a) What is the rotational energy of the system if it is rotated about the z-axis (out of page) with an angular velocity of 5 rad/s

b) What is the rotational energy if the system is rotated about the y-axis?

i-clicker for question b):

A) 281 J B) 291 J C) 331 J D) 491 J E) 582 J

Black board example 10.3

i-clicker

13

2

4

Four small spheres are mounted on the corners of a weightless frame as shown.

M = 5 kg; m = 2 kg;

a = 1.5 m; b = 1 m

Moment of inertia (rotational inertia) of an object depends on:

- the axis about which the object is rotated.- the mass of the object. - the distance between the mass(es) and the axis

of rotation.

2i

ii rmI

Calculation of Moments of inertia for continuous extended objects

dVrdmrmrIi

iimi

222

0lim

Refer to Table10.2

Note that the moments of inertia are different for different axes of rotation (even for the same object)

MLI121

MRI21

MLI31

Moment of inertia for some objects Page 287

Rotational energy earth.The earth has a mass M = 6.0×1024 kg and a radius of R = 6.4×106 m. Its distance

from the sun is d = 1.5×1011 m What is the rotational kinetic energy of

a) its motion around the sun?

b) its rotation about its own axis?

Black board example 10.4

Parallel axis theorem

Rotational inertia for a rotation about an axis that is parallel to an axis through the center of mass

h

2MhII CM CMI

What is the rotational energy of a sphere (mass m = 1 kg, radius R = 1m) that is rotating about an axis 0.5 away from the center with = 2 rad/sec?

Blackboard example 10.5

Conservation of energy (including rotational energy):

finalrotationalfinallinearfinitialrotationalinitiallineari

fi

KKUKKU

EE

,,,,

Again:

If there are no non-conservative forces energy is conserved.

Rotational kinetic energy must be included in energy considerations!

Connected cylinders.

Two masses m1 (5.0 kg) and m2 (10 kg)

are hanging from a pulley of mass M (3.0 kg) and radius R (0.10 m), as shown. There is no slip between the rope and the pulleys.

(a) What will happen when the masses are released?

(b) Find the velocity of the masses after they have fallen a distance of 0.5 m.

(c) What is the angular velocity of the pulley at that moment?

Black board example 10.6

Torque

A force F is acting at an angle f on a lever that is rotating around a pivot point. r is the distance between the pivot point and F.

This force-lever pair results in a torque t on the lever

ft sin Fr

fsinF

fcosF

F

r f

Black board example 10.7

i-clicker

Two mechanics are trying to open a rusty

screw on a ship with a big ol’ wrench.

One pulls at the end of the wrench (r = 1 m)

with a force F = 500 N at an angle F1 = 80°;

the other pulls at the middle of wrench with

the same force and at an angle F2 = 90°.

What is the net torque the two mechanics are applying to the screw?

A. 742 Nm B. 750 Nm C. 900 Nm D. 1040 Nm E. 1051 Nm

Particle of mass m rotating in a circle with radius r.

Radial force Fr to keep particle on circular path.

Tangential force Ft accelerates particle along tangent.

Torque t and

angular acceleration .

tt maF

Torque acting on particle is proportional to angular acceleration : t I

Work in rotational motion:

Definition of work:

Work in linear motion:

sFW

sdFdW

cos

sFsFW

sdFdW

Component of force F along displacement s. Angle between F and s.

tt

WddW

sdFdW Torque t and angular displacement .

Linear motion with constant linear acceleration, a.

tavv xxixf

2

21 tatvxx xxiif

)(222ifxxixf xxavv

tvvxx xfxiif )(21

Rotational motion with constant rotational acceleration, .

tif

2

21 ttiif

)(222ifif

tfiif )(21

Summary: Angular and linear quantities

2

21 IKR

It

Kinetic Energy:

Torque:

Linear motion Rotational motion

2

21 vmK

F ma

Kinetic Energy:

Force:

Momentum: p mv

L I

Angular Momentum:

Work: sFW t WWork:

Superposition principle:

Rolling motion = Pure translation + Pure rotation

Rolling motion

Kinetic energy

of rolling motion:2 21 1

2 2CM CMK Mv I

A ring, a disk and a sphere (equal mass and diameter) are rolling down an incline.

All three start at the same position; which one will be the fastest at the end of the incline?

Black board example 10.8

Demo

A. All the same

B. The disk

C. The ring

D. The sphere