Transcript 7.1 Characteristics for Uniform Circular Motion Objectives

7.1 Characteristics of Uniform Circular Motion
Objectives
1.
2.
3.
4.
Speed and Velocity
Acceleration
The Centripetal Force Requirement
Mathematics of Circular Motion
Homework: castle learning
Uniform circular motion
• Uniform circular motion is the motion of an
object in a circle with a constant or uniform
speed. The velocity is changing because the
direction of motion is changing.
Velocity:
•magnitude: constant
•Direction: changing, tangent
to the circle,
Circular Velocity
Objects traveling
How do we
in circular
define VELOCITY?
motion have constant
speed
and constantly
CHANGINGto
velocity
Velocity
is TANGENT
the –
What
‘d’are
‘t’
are
we
we talking
talkingabout?
about?
changing
in
direction
but
not
magnitude
d
v
circle at all points
t
CIRCUMFERENCE
PERIOD (T)
Time C
for= one
2πr revolution
= πd 2r
vc 
T
Speed (tangential speed) and velocity
Speed, magnitude of velocity:
vavg
2R

T
Direction of velocity:
Tangent to the circle.
vavg =
2·π·R
T
• The average speed and the
Radius of the circle are
directly proportional. The
further away from the center,
the bigger the speed.
Example 1
• A vehicle travels at a constant
speed of 6.0 meters per second
around a horizontal circular
curve with a radius of 24 meters.
The mass of the vehicle is 4.4 ×
103 kilograms. An icy patch is
located at P on the curve. On
the icy patch of pavement, the
frictional force of the vehicle is
zero. Which arrow best
represents the direction of the
vehicle's velocity when it reaches
icy patch P?
a
b
c
d
Example 2
• The diagram shows a 2.0kilogram model airplane attached
to a wire. The airplane is flying
clockwise in a horizontal circle of
radius 20. meters at 30. meters
per second. If the wire breaks
when the airplane is at the
position shown, toward which
point where will the airplane
move? (A, B, C, D)
Example 3
• The diagram shows an object with
a mass of 1.0 kg attached to a
string 0.50 meter long. The object
is moving at a constant speed of
5.0 meters per second in a
horizontal circular path with center
at point O.
• If the string is cut when the object
is at the position shown, draw the
path the object will travel from this
position.
Acceleration
• An object moving in uniform circular motion is moving in a circle
with a uniform or constant speed. The velocity vector is
constant in magnitude but changing in direction.
• Since the velocity is changing. The object is accelerating.
where vi represents the initial velocity and vf represents the final velocity after some
time of t
Example 4
• The initial and final speed of a ball at two different points in time
is shown below. The direction of the ball is indicated by the arrow.
For each case, indicate if there is an acceleration. Explain why or
why not. Indicate the direction of the acceleration.
a.
b.
No, no change in
velocity
yes, change in
velocity, right
c.
yes, change in
velocity, left
d.
yes, change in
velocity, left
Direction of the Acceleration Vector in uniform
circular moiton
+
• The velocity change is directed towards point C - the center of the
circle.
• The acceleration of the object is dependent upon this velocity change
and is in the same direction as this velocity change. The acceleration
is directed towards point C, the center as well, this acceleration is
called the centripetal acceleration.
Example 5
An object is moving in a clockwise direction around a circle at constant speed.
1. Which vector below represents the direction of the velocity vector when the
d
object is located at point B on the circle?
2. Which vector below represents the direction of the acceleration vector when
b
the object is located at point C on the circle?
3. Which vector below represents the direction of the velocity vector when the
a
object is located at point C on the circle?
4. Which vector below represents the direction of the acceleration vector when
the object is located at point A on the circle?
d
The Centripetal Force Requirement
• According to Newton's second law of motion, an
object which experiences an acceleration requires a
net force.
• The direction of the net force is in the same direction
as the acceleration. So for an object moving in a
circle, there must be an inward force acting upon it in
order to cause its inward acceleration. This is
sometimes referred to as the centripetal force
requirement.
• The word centripetal means center seeking. For
object's moving in circular motion, there is a net
force acting towards the center which causes the
object to seek the center.
Centripetal Force
• Inertia causes objects to travel STRAIGHT
• Paths can be bent by FORCES
• CENTRIPETAL FORCE bends an object’s
path into a circle - pulling toward the
CENTER
Inertia, Force and Acceleration for an
Automobile Passenger
• Observe that the passenger (in blue) continues in a straight-line
motion until he strikes the shoulder of the driver (in red). Once
striking the driver, a force is applied to the passenger to force
the passenger to the right and thus complete the turn. This
force is the cause for turning. Without this force, the passenger
would still travel in a straight line at constant speed (inertia).
Inertia, Force and Acceleration
Without a centripetal force, an object in
motion continues along a straight-line
path.
With a centripetal force, an object in
motion will be accelerated and change
its direction.
Centrifugal force is a not real
• centrifugal (center fleeing) force
– A ‘fictitious’ force that is experienced from
INSIDE a circular motion system
• centripetal (center seeking) force
– A true force that pushes or pulls an object
toward the center of a circular path
Example 6
• An object is moving in a clockwise direction around a circle at
constant speed
1. Which vector below represents the direction of the force vector when the object is located
at point A on the circle?
d located
2. Which vector below represents the direction of the force vector when the object is
at point C on the circle?
3. Which vector below represents the direction of the velocity vector when the object is
b
located at point B on the circle?
4. Which vector below represents the direction of the velocity vector when the object is
located at point C on the circle?
d
5. Which vector below represents the direction of the acceleration vector when the object is
located at point B on the circle?
a
c
Mathematics of Uniform Circular Motion
vavg =
a=
v2
R
2·π·R
T
a=
4π2R
T2
Relationship between quantities
a=
v2
R
Fnet and a is directly proportional to the v2.
F ~ v2
a ~ v2
If the speed of the object is doubled, the net force required for that object's
circular motion and its acceleration are quadrupled.
F~m
1
F~
R
a is not related to mass
1
a~
R
Example 1
•
1.
2.
3.
4.
A car going around a curve is acted upon by
a centripetal force, F. If the speed of the car
were twice as great, the centripetal force
necessary to keep it moving in the same
path would be
F
F = m v2 / r
2F
F in directly proportional to v2
F/2
4F
2
F in increased by 2
Example 2
• Anna Litical is practicing a centripetal force demonstration at
home. She fills a bucket with water, ties it to a strong rope,
and spins it in a circle. Anna spins the bucket when it is halffull of water and when it is quarter-full of water. In which case
is more force required to spin the bucket in a circle? Explain
using an equation.
It will require more force to accelerate a half-full bucket of water
compared to a quarter-full bucket. According to the equation Fnet = m•v2 /
R, force and mass are directly proportional. So the greater the mass, the
greater the force.
Example 3
•
1.
2.
3.
4.
The diagram shows a 5.0-kilogram cart traveling
clockwise in a horizontal circle of radius 2.0 meters
at a constant speed of 4.0 meters per second. If
the mass of the cart was doubled, the magnitude
of the centripetal acceleration of the cart would be
doubled
halved
unchanged
quadrupled
ac = v2 / R
Example 4
•
1.
2.
3.
4.
Two masses, A and B, move in circular paths as shown in the
diagram. The centripetal acceleration of mass A, compared
to that of mass B, is
the same
twice as great
one-half as great
four times as great
F = m v2 / r
Example 5
• A 900-kg car moving at 10 m/s takes a turn around a
circle with a radius of 25.0 m. Determine the
acceleration and the net force acting upon the car.
Given: = 900 kg; v = 10.0 m/s R = 25.0 m
Find: a = ? Fnet = ?
a = v2 / R
a = (10.0 m/s)2 / (25.0 m)
a = (100 m2/s2) / (25.0 m)
a = 4 m/s2
Fnet = m • a
Fnet = (900 kg) • (4 m/s2)
Fnet = 3600 N
Example 6
• A 95-kg halfback makes a turn on the football field. The halfback
sweeps out a path which is a portion of a circle with a radius of
12-meters. The halfback makes a quarter of a turn around the
circle in 2.1 seconds. Determine the speed, acceleration and net
force acting upon the halfback.
Given: m = 95.0 kg; R = 12.0 m; Traveled 1/4 of the circumference in 2.1 s
Find: v = ?
a=?
Fnet = ?
a = v2 / R
a = (8.97 m/s)2/(12.0 m)
v=d/t
v = (1/4•2•π•12.0m) /(2.1s)
a = 6.71 m/s2
v = 8.98 m/s
Fnet = m*a
Fnet = (95.0 kg)*(6.71 m/s2)
Fnet = 637 N
Example 7
• Determine the centripetal force acting upon a 40-kg child who
makes 10 revolutions around the Cliffhanger in 29.3 seconds.
The radius of the barrel is 2.90 meters.
Given: m = 40 kg; R = 2.90 m; T = 2.93 s (since 10 cycles takes 29.3 s).
Find: Fnet
Step 1: find speed: v = (2 • π • R) / T = 6.22 m/s.
Step 2: find the acceleration: a = v2 / R = (6.22 m/s)2/(2.90 m) = 13.3 m/s2
Step 3: find net force: Fnet = m • a = (40 kg)(13.3 m/s/s) = 532 N.