Physics 2211: Lecture 17

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Physics 2211 Lecture 17

• Circular Orbits
• Fictitious forces
• Nonuniform Circular Motion

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SEEM to be very different, BUT they have the same
free body diagram . . .
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Orbital Motion is Projectile Motion!
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Orbiting projectile is in free fall
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A Peek Ahead
Our resident genius, Isaac Newton, expressed the
acceleration due to Earths gravity in the form
g GM/R2 where R is the radius of the Earth, M
is the mass of the Earth, and G is a universal
constant. This formula works when the radius ( r
) of the orbit is close to the radius of the
Earth. That is true when h lt lt R. In that case,
Keplers Law
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Orbit Radius R
Johannes Kepler
Orbital Period T
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Fictitious forces in Non-Inertial Frames of
Reference
A car and driver move a constant speed. Suddenly,
the driver brakes
Outside observer if the seat is
frictionless, the driver continues forward at
constant speed and collides with the front
window.
Inside observer a force F throws the driver
against the front window.
F
F is fictitious the observer is in a
non- inertial (accelerated) frame where Newtons
laws do not apply. There is no true push or
pull from anything.
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A car is passing over the top of a hill at speed
v. At this instant,

• n gt w
• n w
• n lt w
• 4 Cant tell without knowing v

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A ball on a string swings in a vertical circle.
The string breaks when the string is horizontal
and the ball is moving straight up. Which
trajectory does the ball follow thereafter?
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Safe Tractor Operation David E.
BakerAgricultural Engineering Extension In
1999 about 780 people in the United States died
in agricultural work accidents. Tractors
accounted for two out of every five accidents.
A tractor turning in a circle at constant speed
is accelerating toward the center of the circle.
If the friction with the seat is negligible, the
tractor driver feels a fictitious centrifugal
force that throws him off the tractor.
Without friction from the seat, there is zero net
force on the driver. Therefore, according to
Newton, he simply moves in a straight line
tangent to the circle. He falls because the
tractor curves out from under him.
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Roller-Coaster
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Parabolic trajectory of water without bucket
Water filled bucket
Critical angular velocity is the angular velocity
where weight force alone is sufficient to cause
circular motion at the top.
Physics is identical to the previous case!
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Full Disclosure
mg
N
N
N
a
mg
mg
a
N
mg
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Non-Uniform Circular Motion
Tangential Acceleration
Radial (centripetal) Acceleration
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Dynamics of Non-Uniform Circular Motion
Just Newtons Second Law . . .
special case at constant
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Non-Uniform Circular Motion constant at
angular acceleration
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R
R
H
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In the previous problem, the tangential
acceleration is constant. What about the radial
acceleration?
The radial acceleration decreases as the particle
slows down.
Does the analysis we have performed apply to a
particle projected with initial speed v0 up a
frictionless circular ramp?
No!
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n
mg
The angular acceleration is not constant as the
motion proceeds. The simple kinematic formulae
cannot be used.
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A block slides around a circle of radius R on a
rough table top. What is the linear speed of the
block after it has traced out ¾ of its circular
trajectory?
V0 7 m/s
R 2m
Top View
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On the motion diagram, the red arrows show the
radial acceleration at four sequential points of
circular motion. On the path from point 2 to
point 3, the average tangential acceleration is
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1

• zero
• positive
• negative
• not enough information to tell

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An airplane is flying in a horizontal circle at a
speed of 482 km/h. The wings of the plane are
tilted at 38.2? to the horizontal. Find the
radius of the circle in which the plane is
flying. Assume that the centripetal force is
provided entirely by the lift force perpendicular
to the wing surface.
1 2.3 km
4 5.3 km
2 3.8 km
5 1.4 km
3 0.5 km
6 none of the above
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Recap
Non-Uniform Circular Motion
Next Lecture
Newtons Third Law Knight Chapter 8