Transcript Chapter 7 - Circular Motion

DoDEA – Physics (SPC 501) - Standards
Pb.9 – Explain how torque () is affected by the
magnitude (F), direction(Sin θ), and point of
application of force (R).  = R*F(Sin θ)
R = Torque Arm = distance between applied force and pivot point
Pb.10 – Explain the relationships among speed,
velocity, acceleration, and force in
rotational systems.
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What is Circular Motion ?
Circular motion (rotation) - When an object
turns about an internal axis
- motion of an object in a circle with a
constant or uniform speed (velocity???)
- constant change in direction = acceleration
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Which “dot” (A-D) is traveling the fastest?
Angular Speed (ω) – the rate at which a body
rotates around an axis (rotations per minute = rpm)
Period - (cycle) – the time it takes to travel one revolution.
-- Object repeatedly finds itself back where it started.
D
C
B
A
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ABCD
SO, which “dot” (A-D) is traveling the fastest?
Radial Distance (r) – (radius) – distance from the axis (center)
Angular Displacement (θ) - the angle through which a point is rotated
C=2πr
D
C
A B
distance = rate  time
distance 2r
time =

rate
v
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Circular Motion is characterized by two kinds of
speeds:
- Tangential (v) (or linear) speed (circumference / t)
- Angular speed (ω) (rotational or circular) (θ / t)
Angular speed (ω) =
“Clock hand speed”
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ABCD
5
Circular Motion—Tangential Speed
Tangential speed (symbol v) - The distance
traveled by a point on the rotating object
divided by the time taken to travel that distance
• Points closer to the circumference (outer edge) have
a higher tangential speed that points closer to the
center.
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Circular Motion – Rotational Speed
• Rotational (angular) speed - (symbol ) - is the
number of rotations or revolutions per unit of time
• All parts of a rigid merry-go-round or clock hand
turn about the axis of rotation in the same amount
of time. – all parts have the same rotational speed
Tangential speed  Radial Distance  Rotational Speed
 = rx
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Rotational and Tangential Speed
CHECK YOUR NEIGHBOR
A ladybug sits halfway between the rotational axis and
the outer edge of the turntable . When the turntable
has a rotational speed of 20 RPM and the bug has a
tangential speed of 2 cm/s, what will be the rotational
and tangential speeds of her friend who sits at the
outer edge?
A.
B.
C.
D.
1 cm/s
2 cm/s
4 cm/s
8 cm/s
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Rotational and Tangential Speed
CHECK YOUR ANSWER
A ladybug sits halfway between the rotational axis and the
outer edge of the turntable . When the turntable has a
rotational speed of 20 RPM and the bug has a tangential
speed of 2 cm/s, what will be the rotational and tangential
speeds of her friend who sits at the outer edge?
A.
B.
C.
D.
1 cm/s
2 cm/s
4 cm/s
8 cm/s
--Rotational speed () of both bugs is the same
(  20 RPM)
--So if the radial distance (r) doubles, tangential
speed (v) will also double.
--Tangential speed (v) is 2 cm/s  2 = 4 cm/s.
 = rx
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Angular Displacement (θ) - the angle through which a point is rotated
Angular Speed (ω) – the rate at which a body rotates around an
axis (rotations per minute = rpm)
Radial Distance (r) (radius) - distance from the central axis (center)
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In circular motion, the direction of any “point” is constantly changing.
Therefore, the velocity (speed with direction) is also constantly changing.
v1
v2
A change in direction
= A change in velocity
= Acceleration a = Δv/t
Acceleration only occurs
when there is a net force
applied to an object a = F/m
In which direction must the FORCE be applied
in order for an object to continue to travel in
constant circular motion?
Direction of Force =
Direction of Acceleration
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Centripetal Acceleration (Ac) – Acceleration directed
toward the center of a circular path
--Always points toward center of circle.
--(Centripetal = center seeking)
--Always changing direction!
NOTE: Velocity is always in a straight line.
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Acceleration only occurs when there is a net force applied
to an object a = F/m
• Any force directed toward a fixed center is
called a centripetal force (Fc). The
magnitude of the force required to maintain
uniform circular motion.
Example: To whirl a tin can at
the end of a string, you pull
the string toward the center
and exert a centripetal
force to keep the can
moving in a circle.
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• Centripetal Force (Fc) Depends upon:
– Mass (m) of object.
– Tangential speed (v) of the object.
– Radius (r) of the circle.
mass  tangential speed
Centripetal force 
radius
F  ma
mv
F
r
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Centripetal Force
2
Centripetal Force—Example
• When a car rounds a
curve, the centripetal force
prevents it from skidding
off the road.
• If the road is wet, or if the
car is going too fast, the
centripetal force is
insufficient to prevent
skidding off the road.
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Centripetal Force
CHECK YOUR NEIGHBOR
Suppose you double the speed at which you round a
bend in the curve, by what factor must the centripetal
force change to prevent you from skidding?
A. Double
B. Four times
C. Half
D. One-quarter
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Centripetal Force
CHECK YOUR ANSWER
Suppose you double the speed (v) at which you round a bend
in the curve, by what factor must the centripetal force (Fc)
change to prevent you from skidding?
A. Double
B. Four times
C. Half
D. One-quarter
Explanation:
mass  tangential speed 2
Centripetal force 
radius
Because the term for “tangential speed” is
squared, if you double the tangential speed,
the centripetal force will be double squared,
which is four times. The radius is constant.
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WARNING: If you make a turn at
double the speed, you are 4 times more
likely to skid off of the road.
Centripetal Force
CHECK YOUR NEIGHBOR
Suppose you take a sharper turn than before and
halve the radius, by what factor will the centripetal
force need to change to prevent skidding?
A. Double
B. Four times
C. Half
D. One-quarter
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Centripetal Force
CHECK YOUR ANSWER
Suppose you take a sharper turn than before and halve the
radius; by what factor will the centripetal force need to change
to prevent skidding? Because the term for “radius” is in the
denominator, if you halve the radius, the
A. Double
centripetal force will double.
B. Four times
2
tangential
speed
mass

C. Half
Centripetal force 
radius
D. One-quarter
WARNING: The sharper your turn
(smaller radius), the MORE likely you are
to skid. The wider your turn (larger
radius), the LESS likely you are to skid.
What applies this centripetal force to the car???
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Direction of Centripetal Force (Fc),
Acceleration (Ac) and Velocity (v)
Without a centripetal
force, an object in
motion continues along
a straight-line path.
Newton’s 1st Law
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An object will change direction
ONLY if there is a NET force
applied to the object. A CONSTANT
force applied towards a central
point results in a circular path
The cork will ONLY move while experiencing
a net force (resulting in an acceleration). If I
were to walk at a constant velocity, the cork
would stay in the center.
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Direction of Centripetal Force (Fc),
Acceleration (Ac) and Velocity (v)
Velocity (v) = straight line
Force (Fc) = toward center
Acceleration (ac) = toward
center
Acceleration occurs in the
same direction as the
applied force
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mv
A change in velocity is due to? F 
2 r
mass  tangential speed
Centripetal force 
radius
(α)
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2
mv
What if the mass decreases? F 
2 r
mass  tangential speed
Centripetal force 
radius
24
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2
mv
What if the radius decreases? F 
2 r
mass  tangential speed
Centripetal force 
radius
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2
A tube is been placed upon the table and
shaped into a three-quarters circle. A golf ball is
pushed into the tube at one end at high speed.
The ball rolls through the tube and exits at the
opposite end. Describe the path of the golf ball
as it exits the tube.
N
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What provides the centripetal force?
a) Tension
b) Gravity
c) Friction
d) Normal Force
Centripetal force is NOT a new “force”. It is simply a way of
quantifying the magnitude of the force required to maintain a
certain speed around a circular path of a certain radius.
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Tension Can Yield a Centripetal Acceleration:
If the person doubles the
speed of the airplane, what
happens to the tension in the
cable?
mv
F = ma 
r
2
Doubling the speed, quadruples the force (i.e. tension)
required to keep the plane in uniform circular motion.
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Friction Can Yield a Centripetal Acceleration:
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Centripetal Force: Question
A car travels at a constant speed
around two curves. Where is the
car most likely to skid? Why?
mv
F = ma 
r
2
Smaller radius: larger force required to
keep it in uniform circular motion.
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Suppose two identical objects go around in horizontal
circles of identical diameter but one object goes around
the circle twice as fast as the other. The force required
to keep the faster object on the circular path is _____
the force required to keep the slower object on the path.
A.
B.
C.
D.
E.
the same as
one fourth of
half of
twice
four times
The answer is E. As the
velocity increases the
centripetal force required
to maintain the circle
increases as the square of
the speed.
mass  tangential speed
Centripetal force 
radius
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2
31
Suppose two identical objects go around in
horizontal circles with the same speed. The
diameter of one circle is half of the diameter of
the other. The force required to keep the object
on the smaller circular path is ___ the force
required to keep the object on the larger path.
A.
B.
C.
D.
E.
the same as
one fourth of
half of
twice
four times
The answer is D. The centripetal force
needed to maintain the circular motion
of an object is inversely proportional to
the radius of the circle. Everybody
knows that it is harder to navigate a
sharp turn than a wide turn.
mass  tangential speed
Centripetal force 
radius
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A.
B.
C.
D.
E.
Suppose two identical objects go around in horizontal
circles of identical diameter and speed but one object has
twice the mass of the other. The force required to keep the
more massive object on the circular path is
the same as
Answer: D.The mass is directly
one fourth of
proportional to centripetal force.
half of
twice
four times
mass  tangential speed
Centripetal force 
radius
2
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Banked Curves
Q: Why exit ramps in highways are banked?
A: To increase the centripetal force for the higher exit speed.
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The Normal Force Can Yield a Centripetal Acceleration:
Engineers have learned to “bank” curves so that cars can
safely travel around the curve without relying on friction
at all to supply the centripetal acceleration.
How many forces are acting on
the car (assuming no friction)?
35
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Banked Curves
Why exit ramps in highways are banked?
FN cosq = mg
Fc = FN sinq = mv2/r
The steeper the bank (q), the less friction is needed
(more Force Normal is applied toward the
center). This is why/how a race car track can
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allow cars to travel at such high speeds.
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Gravity Can Yield a Centripetal Acceleration:
Hubble Space Telescope
orbits at an altitude of 598 km
(height above Earth’s surface).
What is its orbital speed?
mv
F = ma 
r
2
G  m  ME m  v

2
R
R
2
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The ride starts to spin faster and faster and you FEEL
yourself being pushed back against the metal cage. Even
when the ride starts to tilt skyward and you are looking
DOWN at the ground, still you feel yourself almost “forced”
back so that you do NOT fall to your death. EXPLAIN….
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Centrifugal Force (The F-word)
• Although centripetal force is center directed, an
occupant inside a rotating system seems to
experience an outward force. This apparent
outward force is called centrifugal force.
• Centrifugal means “center-fleeing” or “away
from the center.”
(The F-word)
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Centrifugal Force (The F-word)
– A Common Misconception
• It is a common misconception
that a centrifugal force pulls
outward on an object.
• Example:
– If the string breaks, the object
doesn’t move radially outward.
– It continues along its tangent
straight-line path—because no
force acts on it. (Newton’s first
law)
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An inward net force is required to make a turn in a circle. This inward net force
requirement is known as a centripetal force requirement. In the absence of any net
force, an object in motion (such as the passenger) continues in motion in a straight
line at constant speed. This is Newton's first law of motion. While the car begins to
make the turn, the passenger and the seat begin to edge rightward. In a sense, the
car is beginning to slide out from under the passenger. Once striking the driver, the
passenger can now turn with the car and experience some circle-like motion. There is
never any outward force exerted upon the passenger. The passenger is either moving
straight ahead in the absence of a force or moving along a circular path in the
presence of an inward-directed force.
http://www.physicsclassroom.com/mmedia/circmot/rht.cfm
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This path is an inward force - a centripetal force. That is spelled
c-e-n-t-r-i-p-e-t-a-l, with a "p." The other word - centrifugal, with
an "f" - will be considered our forbidden F-word. Simply don't
use it and please don't believe in it.
http://www.physicsclassroom.com/Class/circles/u6l1d2.gif
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Rotating Reference Frames
• Centrifugal force in a rotating reference
frame is a force in its own right – as real as
any other force, e.g. gravity.
• Example:
– The bug at the bottom of the can experiences
a pull toward the bottom of the can.
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Why we use the f-word
• Centrifugal force in a rotating “reference frame”
can be used to examine the simulation of gravity
in space stations of the future.
• By spinning the space station, occupants would
experience a centrifugal force (simulated gravity)
similar to the bug in the can.
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Simulated Gravity
To simulate an acceleration due to gravity, g, which
is 10 m/s2, a space station must:
Have a radius of about 1 km
(i.e. diameter of 2 km).
Rotate at a speed of about
1 revolution per minute.
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Rotational Inertia
• An object rotating about an axis tends to
remain rotating about the same axis at the
same rotational speed unless interfered
with by some external influence.
• The property of an object to resist changes
in its rotational state of motion is called
rotational inertia (symbol I).
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Rotational Inertia
Depends upon
• mass of object.
• distribution of mass
around axis of rotation.
– The greater the distance
between an object’s mass
concentration and the axis,
the greater the rotational
inertia.
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Rotational Inertia
• The greater the rotational inertia, the
harder it is to change its rotational state.
– A tightrope walker carries a long pole that has a high
rotational inertia, so it does not easily rotate.
– Keeps the tightrope walker stable.
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Rotational Inertia
Depends upon the axis
around which it rotates
• Easier to rotate pencil
around an axis passing
through it.
• Harder to rotate it around
vertical axis passing through
center.
• Hardest to rotate it around
vertical axis passing through
the end.
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Rotational Inertia
The rotational inertia depends upon the shape
of the object and its rotational axis.
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Rotational Inertia
CHECK YOUR NEIGHBOR
A hoop and a disk are released from the top of an
incline at the same time. Which one will reach the
bottom first?
A.
B.
C.
D.
Hoop
Disk
Both together
Not enough information
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Rotational Inertia
CHECK YOUR ANSWER
A hoop and a disk are released from the top of an incline at the
same time. Which one will reach the bottom first?
A. Hoop
B. Disk
C. Both together
D. Not enough information
Explanation:
Hoop has larger rotational
inertia, so it will be slower in
gaining speed.
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Torque
• The tendency of a force to cause rotation
is called torque.
• Torque depends upon three factors:
– Magnitude of the force
– The direction in which it acts
– The point at which it is applied on the object
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Torque
• The equation for Torque is
Torque  lever arm  force
• The lever arm depends upon
– where the force is applied.
– the direction in which it acts.
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Torque—Example
• 1st picture: Lever arm is less than length of handle
because of direction of force.
• 2nd picture: Lever arm is equal to length of handle.
• 3rd picture: Lever arm is longer than length of
handle.
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Rotational Inertia
CHECK YOUR NEIGHBOR
Suppose the girl on the left suddenly is handed a bag
of apples weighing 50 N. Where should she sit order to
balance, assuming the boy does not move?
A. 1 m from pivot
B. 1.5 m from pivot
C. 2 m from pivot
D. 2.5 m from pivot
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Rotational Inertia
CHECK YOUR ANSWER
Suppose the girl on the left suddenly is handed a bag of
apples weighing 50 N. Where should she sit in order to
balance, assuming the boy does not move?
A. 1 m from pivot
B. 1.5 m from pivot
C. 2 m from pivot
D. 2.5 m from pivot
Explanation:
She should exert same torque as before.
Torque  lever arm  force
 3 m  250 N
 750 Nm
Torque  new lever arm  force
750 Nm  new lever arm  250N
 New lever arm  750 Nm / 250 N  2.5 m
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Center of Mass and Center of Gravity
• Center of mass is the average position of
all the mass that makes up the object.
• Center of gravity (CG) is the average
position of weight distribution.
– Since weight and mass are proportional,
center of gravity and center of mass usually
refer to the same point of an object.
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Center of Mass and Center of Gravity
To determine the center of gravity,
– suspend the object from a point and draw a
vertical line from suspension point.
– repeat after suspending from another point.
• The center of gravity lies where the two
lines intersect.
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Center of Gravity—Stability
The location of the center of
gravity is important for
stability.
• If we draw a line straight down
from the center of gravity and it
falls inside the base of the object,
it is in stable equilibrium; it will
balance.
• If it falls outside the base, it is
unstable.
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Angular Momentum
• The “inertia of rotation” of rotating objects is
called angular momentum.
– This is analogous to “inertia of motion”, which was
momentum.
• Angular momentum
 rotational inertia  angular velocity
– This is analogous to
Linear momentum  mass  velocity
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Angular Momentum
• For an object that is small compared with the radial
distance to its axis, magnitude of
Angular momentum  mass tangential speed  radius
– This is analogous to magnitude of
Linear momentum  mass  speed
• Examples:
– Whirling ball at the end of a
long string
– Planet going around the Sun
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Angular Momentum
• An external net torque is required to change
the angular momentum of an object.
• Rotational version of Newton’s first law:
– An object or system of objects will maintain
its angular momentum unless acted upon
by an external net torque.
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Angular Momentum
CHECK YOUR NEIGHBOR
Suppose you are swirling a can around and suddenly
decide to pull the rope in halfway; by what factor would
the speed of the can change?
A. Double
B. Four times
C. Half
D. One-quarter
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Angular Momentum
CHECK YOUR ANSWER
Suppose you are swirling a can around and suddenly decide to
pull the rope in halfway, by what factor would the speed of the
can change?
A. Double
B. Four times
C. Half
D. One-quarter
Explanation:
Angular momentum
 mass tangential speed  radius
Angular Momentum is proportional to radius
of the turn.
No external torque acts with inward pull, so
angular momentum is conserved. Half
radius means speed doubles.
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Conservation of Angular Momentum
The law of conservation of angular momentum
states:
If no external net torque acts on a rotating
system, the angular momentum of that
system remains constant.
Analogous to the law of conservation of linear
momentum:
If no external force acts on a system, the total linear
momentum of that system remains constant.
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Conservation of Angular Momentum
Example:
• When the man pulls the weights inward,
his rotational speed increases!
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Angular Momentum
CHECK YOUR NEIGHBOR
Suppose by pulling the weights inward, the rotational
inertia of the man reduces to half its value. By what
factor would his angular velocity change?
A. Double
B. Three times
C. Half
D. One-quarter
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Angular Momentum
CHECK YOUR ANSWER
Suppose by pulling the weights in, if the rotational inertia of the
man decreases to half of his initial rotational inertia, by what
factor would his angular velocity change?
A. Double
B. Three times
C. Half
D. One-quarter
Explanation:
Angular momentum
 rotational inertia  angular velocity
Angular momentum is proportional to
“rotational inertia”.
If you halve the rotational inertia, to keep the
angular momentum constant, the angular
velocity would double.
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Vertical Circular Motion
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