Centripetal Force
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Transcript Centripetal Force
Centripetal Force
Law of Action in Circles
Motion in a circle has a centripetal acceleration.
For every acceleration there is a net force.
There must be a centripetal force.
• Points to the center of the circle
• Magnitude is ma = mv2/r
The centrifugal force that we describe is just inertia.
• It points in the opposite direction – to the outside
• It isn’t a real force
Conical Pendulum
A 200. g mass hung is from a 50. cm string as a
conical pendulum. The period of the pendulum in a
perfect circle is 1.4 s. What is the angle of the
pendulum? What is the tension on the string?
q
FT
Radial Net Force
The mass has a downward
gravitational force, -mg.
There is tension in the string.
• The vertical component must
cancel gravity
FTy = mg
FT = mg / cos q
Tension: FT = mg / cos q = 2.0 N
Centripetal force:
FTr = mg sin q / cos q = mg tan q
q
FT
FT cos q
FT sin q
mg
Acceleration to Velocity
The acceleration and
velocity on a circular path
are related.
a F / m g tan q
a v2 / r
v 2 gr tan q
v gr tan q
q
FT
r
mg tan q
mg
Period of Revolution
The pendulum period is
related to the speed and
radius.
r L sin q
v 2r / T 2L sin q / T
v 2 gr tan q
4 L sin q / T gL sin q sin q / cosq
2 2
2
2
cos q = 0.973
q = 13°
L
FT
r
2
cos q T g / 4 L
2
q
mg tan q
Vertical Curve
A loop-the-loop is a popular
rollercoaster feature.
There are only two forces
acting on the moving car.
• Gravity
• Normal force
FN
There is a centripetal
acceleration due to the loop.
• Not uniform circular motion
Fg
Staying on Track
If the normal force becomes
zero, the coaster will leave
the track in a parabolic
trajectory.
• Projectile motion
At any point there must be
enough velocity to maintain
pressure of the car on the
track.
Fg
Force at the Top
The forces of gravity and the
normal force are both
directed down.
Together these must match
the centripetal force.
The minimum occurs with
almost no normal force.
The maximum is at the
bottom: a = v2 / r.
Fg
FN
Fc FN Fg
mv2 / r FN mg
v
FN r
gr
m
vmin gr
Horizontal Curve
A vehicle on a horizontal curve has a centripetal
acceleration associated with the changing direction.
The curve doesn’t have to be a complete circle.
• There is still a radius (r) associated with the curve
• The force is still Fc = mv2/r directed inward
r
Fc
Curves and Friction
On a turn the force of static friction provides the
centripetal acceleration.
In the force diagram there is no other force acting in
the centripetal direction.
Fc mv 2 / r
Fc F f
r
Fc
F f s mg
Skidding
The limit of steering in a
curve occurs when the
centripetal acceleration
equals the maximum static
friction.
2
Fc mvmax
/ r F f s mg
v
2
max
/ r s g
vmax s gr
2
r vmax
/ s g
A curve on a dry road (s =
1.0) is safe at a speed of 90
km/h.
What is the safe speed on
the same curve with ice (s =
0.2)?
•
•
•
•
90 km/h = 25 m/s
rdry = v2/ s g = 64 m
v2icy = s g r = 120 m2/s2
vicy = 11 m/s = 40 km/h
Banking
Fc FN sin q F f
2
mvmax
/ r mg tan q s mg
vmax ( s tan q ) gr
Curves intended for
higher speeds are
banked.
Without friction a curve
banked at an angle q can
supply a centripetal force
Fc = mg tan q.
The car can turn without
any friction.
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