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Chapter 4 Lecture
Week 12 Day 1
Circular Motion
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Circular Motion
• Why do pilots sometimes black out while pulling
out at the bottom of a power dive?
• Are astronauts really "weightless" while in orbit?
• Why do you tend to slide across the car seat
when the car makes a sharp turn?
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Centrifugal Force
•
If the car you are in turns a
corner quickly, you feel
“thrown” against the door.
•
The “force” that seems to
push an object to the
outside of a circle is called
the centrifugal force.
• This is caused by your
inertia pressing you against
the door
Fictitious Forces
• If you are riding in a car that makes a sudden stop,
you may feel as if a force “throws” you forward
toward the windshield.
• There really is no such force.
• Nonetheless, the fact that you seem to be hurled
forward relative to the car is a very real experience!
• You can describe your experience in terms of what
are called fictitious forces or pseudo forces. .
Reading Quiz
1. For uniform circular motion, the acceleration
A.
B.
C.
D.
E.
is parallel to the velocity.
is directed toward the center of the circle.
is larger for a larger orbit at the same speed.
is always due to gravity.
is always negative.
Slide 6-6
Answer
1. For uniform circular motion, the acceleration
A.
B.
C.
D.
E.
is parallel to the velocity.
is directed toward the center of the circle.
is larger for a larger orbit at the same speed.
is always due to gravity.
is always negative.
Slide 6-7
Reading Quiz
2. When a car turns a corner on a level road, which force provides
the necessary centripetal acceleration?
A.
B.
C.
D.
E.
Friction
Tension
Normal force
Air resistance
Gravity
Slide 6-8
Answer
2. When a car turns a corner on a level road, which force provides
the necessary centripetal acceleration?
A.
B.
C.
D.
E.
Friction
Tension
Normal force
Air resistance
Gravity
Slide 6-9
Radial acceleration: An object at constant
speed v on a circular path of radius r
• Based on our testing experiments, we found:
• The constant of proportionality is 1, so we can define the
radial acceleration as:
• The acceleration points toward the center.
• The SI units are m/s2.
• In the limiting case of a straight line, the radius goes to
infinity and the acceleration goes to zero. This equation
makes sense.
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Every quantity we studied has a rotational analog
We have mainly discussed particle motion
• This describes motion of the center of mass
• What about rotational motion about the center of mass
• Every motion quantity we have looked at in kinematics, Forces,
momentum, and Energy has an analogous quantity in Rotation.
• Every equation has an analog too
• Rotational equations look exactly the same
• Examples
• Position => theta
• Velocity => angular velocity
• Force => torque
• Newton’s 2nd Law => Newton’s 2nd Law for rotational motion
• Mass => Moment of Inertia
Slide 15-37
Units of rotational position
• The unit for rotational position is the radian (rad).
It is defined in terms of:
– The arc length s
– The radius r of the circle
• The angle in units of radians is
the ratio of s and r:
• The radian unit has no dimensions; it is the ratio
of two lengths. The unit rad is just a reminder
that we are using radians for angles.
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Tip
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Rotational (angular) velocity ω
• Translational velocity is the
rate of change of linear
position.
• We define the rotational
(angular) velocity v of a rigid
body as the rate of change
of each point's rotational
position.
– All points on the rigid
body rotate through the
same angle in the same
time, so each point has
the same rotational
velocity.
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Rotational (angular) velocity ω
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Tips
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Rotational (angular) acceleration α
• Translational acceleration describes an object's
change in velocity for linear motion.
– We could apply the same idea to the center of
mass of a rigid body that is moving as a
whole from one position to another.
• The rate of change of the rigid body's rotational
velocity is its rotational acceleration.
– When the rotation rate of a rigid body
increases or decreases, it has a nonzero
rotational acceleration.
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Rotational (angular) acceleration α
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Tip
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Torque τ produced by a force
• The SI unit of force is the Newton-meter (N-m).
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Tip
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Tip
• To decide the sign of the torque, pretend that a pencil is
the rigid body. Hold it with two fingers at a place that
represents the axis of rotation, and exert a force on the
pencil representing the force whose torque sign you wish
to determine. Does the force cause the pencil to turn
counterclockwise (a positive torque) or clockwise (a
negative torque) about the axis of rotation?
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Conditions of equilibrium
• An object remains at rest with respect to a
particular observer in an inertial reference frame.
• In this chapter, we consider only observers who
are at rest with respect to Earth.
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Condition 2: Rotational (torque) condition of
equilibrium
• A rigid body is in turning or rotational static
equilibrium if it is at rest with respect to the
observer and the sum of the torques (positive
counterclockwise torques and negative
clockwise torques) about any axis of rotation
produced by the forces exerted on the object is
zero.
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Period
• Period is the time interval it takes an object to
travel around an entire circular path one time.
• Period has units of time, so the SI unit is s.
• For constant-speed circular motion, we divide the
distance traveled in one period (the circumference
of the circular path, 2πr) by the time interval T (its
period) to get:
• Do not confuse the symbol T for period with the
symbol T for the tension force.
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Quantitative Exercise 4.3: Singapore hotel
• What is your radial acceleration when you sleep in a hotel
in Singapore at Earth's equator?
• Remember that Earth turns on its axis once every 24
hours, and everything on the planet's surface actually
undergoes constant-speed circular motion with a period
of 24 hours.
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Quantitative Exercise 4.3: Singapore hotel
• We want to determine your radial acceleration when you
sleep in a hotel in Singapore at Earth's equator.
– Earth turns on its axis once every 24 hours, and
everything on its surface actually undergoes
constant-speed circular motion with a period of
24 hours.
• Represent this situation mathematically, THEN solve and
evaluate.
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Is Earth a noninertial reference frame?
• Newton's laws are valid only for observers in
inertial reference frames (nonaccelerating
observers).
– Observers on Earth's surface are accelerating
due to Earth's rotation.
• Does this mean that Newton's laws do not apply?
– The acceleration due to Earth's rotation is
much smaller than the accelerations we
experience from other types of motion.
• In most situations, we can assume that Earth is
not rotating and, therefore, does count as an
inertial reference frame.
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Circular motion component form of
Newton's second law
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Problem-solving strategy: Processes
involving constant-speed circular motion
• Sketch and translate
– Sketch the situation described in the problem
statement. Label it with all relevant information.
– Choose a system object and a specific position
to analyze its motion.
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Problem-solving strategy: Processes
involving constant-speed circular motion
• Simplify and diagram
– Decide if the system can be modeled as a point-like
object.
– Determine if the constant-speed circular motion
approach is appropriate.
– Indicate with an arrow the direction of the object's
acceleration as it passes the chosen position.
– Draw a force diagram for the system object at the
instant it passes that position.
– On the force diagram, draw an axis in the radial
direction toward the center of the circle.
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Problem-solving strategy: Processes
involving constant-speed circular motion
(Cont'd)
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Problem-solving strategy: Processes
involving constant-circular motion
• Represent mathematically
– Convert the force diagram into the radial r-component
form of Newton's second law.
– For objects moving in a horizontal circle (unlike this
example), you may also need to apply a vertical
y-component form of Newton's second law.
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Problem-solving strategy: Processes
involving constant-speed circular motion
• Solve and evaluate
– Solve the equations formulated in the
previous two steps.
– Evaluate the results to see if they are
reasonable (e.g., the magnitude of the
answer, its units, limiting cases).
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Example 4.5: Toy airplane
• A toy airplane flies around in a horizontal circle at constant
speed. The airplane is attached to the end of a 46-cm
string, which makes a 25° angle relative to the horizontal
while the airplane is flying. A scale at the top of the string
measures the force that the string exerts on the airplane.
• Predict the period of the airplane's motion (the time
interval for it to complete one circle).
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Example 4.6: Rotor ride
• A 62-kg woman is a passenger in a rotor ride. A drum of
radius 2.0 m rotates at a period of 1.7 s. When the drum
reaches this turning rate, the floor drops away but the
woman does not slide down the wall. Imagine that you
were one of the engineers who designed this ride.
• Which characteristics would ensure that the woman
remained stuck to the wall?
• Justify your answer quantitatively.
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Loop the Loop
Consider a ball on a string making a vertical circle.
• Draw a free-body diagram of the ball at the top and
bottom of the circle
• Rank the forces in the two diagrams. Be sure to
explain the reasoning behind your rankings
• Find the minimum speed of the ball at the top of the
circle so that it keeps moving along the circular path
• What would happen if the speed was less than the
minimum?
• What would happen if the speed was more than
the miniumum?
Slide 6-12
Example Problem: Loop-the-Loop
A roller coaster car goes through a vertical loop at a constant
speed. For positions A to E, rank order the:
• centripetal acceleration
• normal force
• apparent weight
Slide 6-32
Keep the Water in the Bucket
Slide 6-12
Roller Coaster and Circular Motion
A roller-coaster car has a mass of 500 kg when fully loaded
with passengers as shown on the right.
1. If the car has a speed of 20.0 m/s at point A,
what is the force exerted by the track at this point?
What is the apparent weight of the person?
2. What is the maximum speed the car can have at point
B and stay on the track?
Slide 15-37
Example 4.7: Texas Motor Speedway
• Texas Motor Speedway is a 2.4-km (1.5-mile)-long oval
track. One of its turns is about 200 m in radius and is
banked at 24° above the horizontal.
• How fast would a car have to move so that no friction is
needed to prevent it from sliding sideways off the
raceway (into the infield or off the track)?
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Tip for circular motion
• There is no special force that causes the radial
acceleration of an object moving at constant
speed along a circular path.
• This acceleration is caused by all of the forces
exerted on the system object by other objects.
• Add the radial components of these regular
forces.
• This sum is what causes the radial acceleration
of the system object.
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Summary
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Summary
© 2014 Pearson Education, Inc.