Transcript rotational motion & law of gravity

CIRCULAR MOTION
ROTATIONAL MOTION
• Objects that spin undergo rotational
motion.
• Any point on the object has circular
motion around the axis.
• The direction of motion is constantly
changing.
ROTATIONAL QUANTITIES
•  - angular displacement –(degrees,
radians, or revolutions)
•  = s/r
•  - angular speed (rads/sec)
•  = /t
•  - angular acceleration (rads/sec2)
•  = /t
1 radian = 57.3o
2 rads = 1 rev = 360o
Practice Problem
• While riding on a carousel that is rotating
clockwise, a child travels through an arc
length of 11.5 m. If the child’s angular
displacement is 165o, what is the radius of
the carousel?
Practice Problem
• A child at an ice cream parlor spins on a
stool. The child turns counterclockwise
with an average angular speed of 4.0
rads/sec. In what time interval will the
child’s feet have an angular displacement
of 8.0 rad?
Practice Problem
• The wheel on an upside down bicycle
moves through 11.0 rad in 2.0 s. What is
the wheel’s angular acceleration of its
initial angular speed is 2.0 rad/s?
Tangential Speed
• Instantaneous linear speed
• Varies with position from axis of
rotation
• Speed along a line drawn tangent to
the circular path
vt = r
vt = 2r/T
T is period (time/# revolutions)
Tangential Acceleration
• Tangent to circular path
• Occurs when rotating objects change
speed
• Example: A carousel speeds up
at = r
Practice Problem
• What is the tangential speed of a child
seated 1.2 m from the center of a rotating
merry go round that makes one complete
revolution in 4.0 s?
• It takes 2.5 s for the merry go round to
slow to a speed of .75 m/s. What is the
tangential acceleration?
Centripetal Acceleration
• Occurs as object moves in a circular
path because it changes direction
• Is constant or uniform
• Directed toward the center
ac = vt2
r
ac = r2
Total Acceleration
• When both centripetal
and tangential
acceleration exist, at is
tangent to circular
path
• ac is toward the
center
• Components are
perpendicular
• atotal = square root of
ac2 + at2
• Direction = tan = ac/at
ac
at
atotal
Practice Problem
• The Polar Express has an angular
acceleration of .50 rad/s/s. A rider sits 6.6
m from the center and makes 10
revolutions in 13 seconds. Find the
tangential, centripetal, and total
accelerations.
Circular Motion
• object moves in a circular path
• Continuous uniform acceleration
• Ex: ball on the end of a string, Moon moving
about the Earth (almost circular)
• Can be vertical or horizontal
animation
by Behrooz
Mostafavi
Vertical Circular motion
Vmin occurs at
the top
V = square
root of rg
Ft
Fw
Ex: loop roller
coaster
Ft
Fw
Practice Problem
• A ball of mass .45 kg is swung in a vertical
circle. If the centripetal force on the ball is
12.5 N, what is the tension in the string at
the top and bottom of the circle?
How do you feel…
• when sitting on the outside of the Polar
Express ride at the State Fair
• when you are in a car that turns sharply to
the left.
What causes these feelings?
Centrifugal Force – not real
Centripetal Force – the real
force
• Explain how a bucket of water can be
whirled in a a vertical circle without the
water spilling out, even at the top of the
circle when the bucket is upside down.
Horizontal Circular motion
Tension force acts horizontal and is
constant.
If weight is small enough it can be
ignored.
Ex: polar express
Centrifugal Force
the fake force
What will happen…
• when a car doesn’t have enough friction
force to get around a curve?
• If centripetal
force is
inward why
will water not
fall our of a
cup that is
swung in a
vertical path?
What is required to cause the ball
to have a curved path?
Centripetal Force
• Required to maintain
centripetal
acceleration (Newton’s
Laws)
• Directed toward the
center
• Acts at right angles to
motion
• Ex: gravity, friction,
strings…
Fc = mac = mr2 =
mvt2/r
Practice Problem
• What would be the centripetal force on a
1500 kg car rounding a 8.5 m curve at a
speed of 10.0 m/s?
It is sometimes
said
that water is
removed
from clothes in the
spin
cycle by centrifugal
force
throwing the water
outward.
Is this correct?
The smaller the velocity of the object, the less centripetal
force you will have to apply.
The smaller the length of rope (radius), the more centripetal
force you will have to apply to the rope
.
The smaller the mass, the smaller the centripetal force
(shown by the red vector labeled as the force of tension in
the rope, FT) you will have to apply to the rope.
If you let go of the
rope (or the rope
breaks) the object
will no longer be kept
in that circular path
and it will be free to
fly off on a tangent.
Newton’s Law of Gravitation
• Planets move in nearly circular orbits
about the sun
• Gravity acts as the centripetal force.
• Any two masses are attracted
• Inverse square law
Fg = Gm1m2
r2
G = 6.67 x 10-11 Nm2/kg2
Practice Problem
• What is the force of attraction between you
and the Earth?
Pictures and animations from
• http://regentsprep.org/Regents/physics/ph
ys06/bcentrif/default.htm
• http://www.ap.smu.ca/demos/content/mec
hanics/waiters_tray/waiters_tray.html