nazarin b. nordin [email protected] what you will learn: centripetal force: acceleration,...
TRANSCRIPT
What you will learn:
• Centripetal force: acceleration, centrifugal force/ acceleration, mass-radius polygons
• Centrifugal force applied to wheel balancing/ clutches, governors
• Curved tracks: vehicles overturning/ sliding on level track, vehicles on banked track
• Angular displacement ( q ) is
usually expressed in radians, in
degrees, or in revolutions.
ANGULAR DISPLACEMENT
57.30
One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.
1 rev = 3600 = 2p radians (rad)
1
2
3
4
56
6 segments getsto here.
2p segments gets completely around.
Thus the angle q in radians is given in terms of the arc length l it subtends on a circle of radius r by
r
l
The radian measure of an angle is a dimensionless number.
THE ANGULAR SPEED
The angular speed ( w ) of an object whose axis of
rotation is fixed is the rate at which its angular
coordinate, the angular displacement q, changes
with time. If q changes from qi to qf in a time t,
then the average angular speed is
tif
w = 2 p f.
• f is the frequency in revolutions per second, rotations per second, or cycles per second.
• Accordingly, w is called the angular frequency. We can associate a direction with w and thereby create a vector quantity.
• The units of are exclusively rad/s. Since each complete turn or cycle of a revolving system carries it through 2p rad
THE ANGULAR ACCELERATION• The angular acceleration ( a ) of an object whose
axis of rotation is fixed is the rate at which its
angular speed changes with time.
• If the angular speed changes uniformly from wi
to wf in the time t, then the angular acceleration
is constant and
tif
The units of a are typically rad/s2, rev/min2, and
such.
It is possible to associate a direction with w, and
therefore with a, thereby specifying the angular
acceleration vector a, but we will have no need
to do so here.
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Equations for uniformly accelerated angular motion are exactly analogous to those for uniformly accelerated linear motion. In the usual notation we have:
2
2
22
221
if
i
if
fi
tt
t
t
RELATIONS BETWEEN ANGULAR AND TANGENTIAL QUANTITIES:
• When a wheel of radius r rotates about an axis whose direction is fixed, a point on the rim of the wheel is described in terms of the circumferential distance l it has moved, its tangential speed v, and its tangential acceleration aT.
• These quantities are related to the angular quantities q, w, and a, which describe the rotation of the wheel, through the relations:
rarvrl T
• provided radian measure is used for q, w, and a. • By simple reasoning, l can be shown to be the
length of belt wound on the wheel or the distance the wheel would roll (without slipping) if free to do so.
• In such cases, v and aT refer to the tangential speed and acceleration of a point on the belt or of the center of the wheel.
rarvrl T
An object moving in a circle with constant speed, v, experiences a centripetal acceleration with: *a magnitude that is constant in time and
is equal to
*a direction that changes continuously in time and
always points toward thecenter of the circular path
Uniform Circular Motion
r
va
2
For uniform circular motion, the velocity is tangential to the circle and perpendicular to the acceleration
A circular motion is described in terms of the period T, which is the time for an object to complete one revolution.
Period and Frequency
2r
f 1T
T 2rv
The distance traveled in one revolution is
The frequency, f, counts the number of revolutions per unit time.
r
The moon’s nearly circular orbit about the earth has a radius of about 384,000 km and a period T of 27.3 days. Determine the acceleration of the Moon towards the Earth.
Example of Uniform Circular Motion
T 2rv
v 2rT
a v2
r4 2r2
T 2r4 2rT 2
a 2.7210 3m / s2g
9.8m / s2
2.7810 4g
Uniform Circular MotionNewton’s 2nd Law: The net force on a body is equal to the product of the mass of the body and the acceleration of the body.
*The centripetal acceleration is caused by a centripetal force that is directed towards the center of the circle.
F ma mv2
r
ROTATIONAL INERTIA
• Law of inertia for rotating systemsAn object rotating about an axis tends to remain rotating at the same rate about the same axis unless interfered with by some external influence.
• Examples: bullet, arrow, and earth
• Demo – Football and spinning basketball• Demo - Whirly Tube (Zinger)• Demo – Whirly Shooter• Demo - Disc Gun• Demo - Rubber Bands
• Demo - Inertia Bars
• Moment of inertia (rotational inertia)
The sluggishness of an object to changes in
its state of rotational motion
• Distribution of mass is the key.
• Example: Tightrope walker
CENTRIPETAL ACCELERATION• Centripetal acceleration (ac):• A point mass m moving with constant speed v around
a circle of radius r is undergoing acceleration. • The direction of the velocity is continually changing.• This gives rise to an acceleration ac of the mass,
directed toward the center of the circle. • We call this acceleration the centripetal acceleration;
its magnitude is given by
r
vaC
2
Because v = rw, we also have
r
vac
2
where w must be in rad/s.
r
r 2)( 2r
r
r 22
THE CENTRIPETAL FORCEThe centripetal force (Fc) is the force that must act on a mass m moving in a circular path of radius r to give it the centripetal acceleration v2/r. From F = ma, we have
2222
mr
r
rm
r
vmmaFC
Where Fc is directed toward the center of the circular path.
• Centripetal force - center seeking force
• Examples: tin can and string, sling, moon and
earth, car on circular path
CENTRIPETAL FORCE
• Demo - Coin on clothes hanger
• Demo - String, ball, and tube
• Demo - Loop the loop
CENTRIFUGAL FORCE
• Centrifugal force - center fleeing force
• Often confused with centripetal
• Examples: sling and bug in can
• Demo - Walk the Line
• Centrifugal force is attributed to inertia.
CENTRIFUGAL FORCE IN A ROTATING REFERENCE FRAME
• A frame of reference can influence our view
of nature.
• For example: we observe a centrifugal
force in a rotating frame of reference, yet it
is a fictitious (pseudo) force.
• Centrifugal force stands alone (there is no
action-reaction pair) - it is a fictitious force.
• Another pseudo force - Coriolis
THANK YOU