Download vertical circles banked curves

Circular Motion
• Accelerated requires
Fnet.
• Acceleration called ac.
• Fnet called centripetal
force.
The term “Centripetal Force” decribes the
direction of Fnet & acceleration. It is NOT itself
a force.
• Any force applied at 90o to displacement or forces
can cause curved or circular motion.
• Fc is the amount of force required to keep an
object of mass m, moving at speed v, in a circle of
radius r.
Circular or Curved Motion
Equations
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•
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•
T = time to complete a cycle (sec)
f = number of cycles in unit time (hz)
T = 1/f
vc = d/t
= 2pr/T
ac = v2/r
Fc = mac = mv2/r.
Centripetal force and acceleration
may be caused by:
• gravity • friction –
• Tension in a rope or cord• Felc –
• F
mag
• R (Fn) –
• Etc.
1. Write an equation for centripetal
acceleration in terms of r and T.
Substituting 2pr for v in
T
accl eq. a = v2
r
ac = 4p2r
T2.
2. The distance from moon’s center to
earth’s center = 3.8 x 108 m.
Moon’s ac = 2.8 x 10-3 m/s2. What is
moon’s period?
• 2.3 x 106 seconds
• ~ 27 days.
The car is turning due to the inward force,
you feel as though you are being forced
leftward or outward. The car is beginning
its turning motion (to the right) while you
continue in a straight line path.
Direction
Velocity Tangent to circular path.
Accl & force toward center of circular path.
Mud sticks to tire.
Circular or Curved Motion Equations
•
•
•
•
•
•
•
T = time to complete a cycle (sec)
f = number of cycles in unit time (hz)
T = 1/f
vc = d/t
= 2pr/T
ac = v2/r
Fc = mac = mv2/r.
Sometimes a measured in g’s.
Multiples of Earth’s a of gravity.
1g = 9.81 m/s2.
2g = 19.6 m/s2.
3g = 29.4 m/s2.
.
.
.
etc.
An 80-kg astronaut experience a force of 2890N when orbiting Earth. How many g’s does he
feel?
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ac = F/m
2890 N / 80-kg = 36 m/s2.
36 m/s2 / 9.81 m/s2
= 3.7 g.
Vertical Circles
Swing keys in vertical circle. Gravity Fg
acts vertically.
Sketch Free body diagrams top and
bottom.
What about a cart on a track?
At the top, Fc is combo of weight & Fn.
Write the Fnet equation. Take center as +.
Fn
W
Fnet = Fc = mv2/r = mg + Fn.
Minimum v occurs just as Fn = 0, when you
lose contact with your seat, you begin to
fall.
Fc = mv2/r = mg + Fn
When Fn = 0,
mv2/r = mg. So mass is irrelevant.
Rearrange, solve for vmin,
v = (gr)1/2. v depends on radius only.
R (FN) may also be called "apparent weight" this
is the force of the seat on the rider and also
describes what the rider "feels" .
If R is less than the rider's weight, rider will
fall out. R might become negative, a safety
restraint system -- seat belts, lap bars,
shoulder restraints, are needed.
Tension in Ropes
Swinging Keys
To find the required tension at the top of the
arc: Write a free body Fnet expression:
Fc = T + mg.
Fc – mg = T
mv2/r – mg = T.
The tension is equivalent to R in the ride.
At the bottom the vel is max & Fc = T – W.
T.
Fc = T – mg.
mv2/r = T – mg.
W.
The necessary tension
to swing in a circle is: mv2/r + mg = T.
Ex 1: A 10 kg package is swinging in a
vertical circle of radius 5.0 m at the end
of a rope. If the ac is 5.0 m/s2,
a. Sketch the free body diagram at
the bottom of the loop.
b. what is the tension in the rope
when the package is at the bottom
of the loop?
• Fc = T – mg.
•
•
•
mv2/r + mg = T.
m (v2/r + g) = T.
m(ac + g) = T.
T.
mg.
• 10kg (5.0 m/s2 + 10 m/s2.)
• 150 N.
Ex 2. A motorcyclist rides in a vertical
circle of 20.0 m radius. What is the
minimum speed he must have at the
top to complete the loop?
14 m/s.
• Hwk Packet.
Banked Curves
Wall of Death
• https://www.youtube.com/watch?v=QA1w
wR9PGxU
Banked Curves.
Flat road only friction acts supplies Fc to pull car into curves.
Banked curves add R component toward the center.
If the curve is banked, there’s a critical speed vc, Ff =0, the car
can turn without slipping out.
Free Body Diagram Banked Curve
Normal Force Components
sinq = Fc
n
Fc
tan q = Fc
mg
This is NOT the
same vector
sketch as for Fll.
Fc = mac points horizontally toward the
center of the curve.
This component of the normal is supplying
the Fc to keep the car moving through the
banked curve.
If friction is present both the
horizontal component of N and the
Friction force contribute to Fc.
FN tan θ + Ff = Fc.
Read Hamper (purple book) pg 40- 42.
Do handout problems Ex 2.4 prb 1 – 5 and pg 143
#3.
IB Problem Packet Banked Curve
• http://www.youtube.com/watch?v=QmAd
fLlhfzw