Uniform Circular Motion AP - Linville UCM AP.pdf · • Uniform circular motion is motion in a...
Transcript of Uniform Circular Motion AP - Linville UCM AP.pdf · • Uniform circular motion is motion in a...
• Uniform circular
motion is motion
in a circle at the
same speed
• Speed is
constant, velocity
direction
changes
• the speed of an object moving in a circle is given
by
2 r
Tv
circumference
time
r= radius of
circle
T = period, time for 1
complete circle
Moon setting
over Rio de
Janerio
Pictures every
6.5 minutes
showing the
moon is moving
at a uniform
speed
Centripetal Acceleration
• The direction of motion
is changing so the
object is accelerating
t
vv
t
va
if
• the centripetal acceleration of the object is
r
2vc
a ac = centripetal acceleration
(always to centre of circle)
•the velocity vector is
always directed along
the tangent of the circle
(at right angles to the
acceleration vector)
•centripetal means
centre-seeking, the
acceleration vector
always points to the
centre of the circle
•Acceleration is radially
inward
Centripetal Force
• inertia tends to keep objects moving in a
straight line, a force is needed to cause a
circular motion (2nd Law of Motion)
• the force causing the motion is called the
centripetal force and it always acts
towards the centre of the circle (parallel to
the acceleration vector)
• an object will move in a circle whenever a
constant magnitude force acts on the
object at right angles to its direction of
motion
• The string pulls ball
towards the centre
of the circle
• 3rd Law of Motion
states that a reaction
force will act
outward (ball pulls
string)
• this is sometimes
called centrifugal
force
Important!!!
• centripetal force is not a fundamental force
like gravity, it is the net force acting on an
object moving in a curved path
• The force will not change the speed of the
object because the force has NO
component parallel to the velocity
vector
Example
• A centrifuge for pilot training is 11 m in
radius. At what speed must it rotate in
order to inflict 7.0 g on a pilot?
Frequency
• sometimes speed is given in revolutions per minute (rpm) or revolutions per second (rps)
• this is how many times the object goes around the circle in 1 minute or 1 second
• frequency is the number of cycles per second measured in hertz (Hz)
Free Body Diagrams
• A tetherball is attached to a
swivel in the ceiling by a light
cord. When the ball is hit by a
paddle, it swings in a horizontal
circle with constant speed, and
the cord makes a constant
angle with the vertical
direction. Write the expression
for the centripetal force in
terms of the other forces.
Free Body Diagrams
• An object is on a horizontal disc that is
rotating at constant speed. Friction
prevents the rock from sliding. Write the
expression for the centripetal force in
terms of the other forces.
•
Example
• A 3.50 kg object is swung in a 1.50 m radius
horizontal circle at 40.0 rps (40.0 Hz). What
magnitude force acts on the object?
2
2
4 mF m a F net c
T
r
Effect of radius on speed and forces
• all points on a rotating solid
have the same period, but
different speeds
• because the inner points
have smaller distances to
travel, their speeds are less
• the speed and force depends
on the radius
Rotating
disc
• NASA and other space agencies use this to help launch satellites
• at the equator, the speed is about 1667 km/h, at Edmonton, it is about 900 km/h
Angular displacement & velocity
• The rotational
displacement of a
point, , in radians
• 2 radians = 360o
• Angular velocity,
(lower case omega)
radians/s
• Every point on the
disc has the same
angular velocity
avet
• By convention, the angular displacement
is positive if it is counterclockwise and
negative if it is clockwise.
Example: Adjacent Satellites
Two satellites are put into
an orbit whose radius is
4.23×107m.
If the angular separation of the two
satellites is 2.00 degrees, find the
arc length that separates them.
rad 0349.0deg360
rad 2deg00.2
r
s
Radius
length Arcradians)(in
s = r = 4.23x107 m x 0.0349 rad
s = 1.48 x 106 m
Rolling Motion
• A rolling wheel has
rotational and
translational velocities
• (a) axle is at rest and
the wheel is rotating
at v = r
Rolling Motion
• A rolling wheel has
rotational and
translational velocities
• (b) the wheel is
moving at v = r,
every point on the
wheel is moving
horizontally (wheel is
skidding)
Example
• A diver completes 1.5
rotations in 2.3 s.
Determine the
average angular
speed of the diver
Solution
• =1.5 rotations x 2 rad
= 9.425 radians
4.1 /ave rad st
Example
• A metal cylinder of radius 0.45 m is
spinning at 2000 rpm and a brake is
applied slowing it to 1000 rpm in 10
seconds. What is the angular
acceleration?
Solution
• In one revolution there are 2 radians
• (2000 rev/min) x (2 rad/sec) x (1 min/60
sec) = 209.4 rad/sec
• (1000 rev/min) x (2 rad/sec) x (1 min/60
sec) = 104.7 rad/sec
Example
• A bullet is fired
through 2 discs
rotating at 99.0 rad/s.
The discs are 0.955
m apart. The angular
displacement
between the holes is
0.260 rad. Calculate
the speed of the
bullet.