Download 6.3 Apparent forces in circular motion

Section 6.3 Apparent Forces in Circular Motion NOTES
AP Physics I
Apparent Forces in Circular Motion
As an object turns, another body inside that object but not connected to it will want
to keep moving straight by inertia (Newton’s First Law). The result is that the
object runs into the body. To the body, it feels that there is an outward force
acting, but this force, called Centrifugal force, does not exist.
If an open container of water is swung fast enough in a circle the water will not
leave the container. Also, a car of a roller coaster doing a loop does not fall off the
track. It is important to analyze the forces that make these situations possible.
The figure below represents a roller coaster car going around a loop with a radius
x (x points toward center)
of r.
v
n
w
Fnet
Fnet
r
n
x (x points toward center)
Fnet
Fnet
w
v
At the bottom of the loop, the only forces acting are the weight of the person and
the normal force of the seat pushing up on the person. The force you feel, your
apparent weight, is the magnitude of the contact force that supports you. At the
bottom, the seat is supporting the person with the normal force, so the apparent
weight is wapp = n. From our understanding of circular motion, we can say:
 The person is moving in a circle, so there must be a net force
directed toward the circle’s center to give centripetal acceleration.
 The net force points upward, so n > w.
 The apparent weight is wapp = n, which must be larger than the true
weight (since n > w).
The sum of the forces in the x direction (with x taken as toward the center of the
mv 2
circle) are nx + wx = n – w =
. The apparent weight is:
r
wapp
mv 2
.
nw
r
At the bottom, then, the apparent weight is greater than the true weight.
At the top of the loop the normal force is pushing down. There must be a net force
downward, toward the circle’s center. If x is positive toward the center, then the
sum of the forces is:
mv 2
 Fx  nx  wx  n  w  r
The apparent weight, n, is now
wapp
mv 2
n
w
r
Notice that if the velocity is large enough, the apparent weight can be larger than
the true weight. However, if v is low enough a condition might arrive where
mv 2
 w . At this condition, the apparent weight becomes zero, and the seat does
r
not push the person. With this situation, the person can complete the circle only
due to the weight force supporting the centripetal acceleration. The speed where n
= 0 is called the critical speed, vc. For this case, mvc2/r = w, so the critical speed is:
vc 
rw
rmg

 gr
m
m
If the speed is less than the critical speed the apparent weight would be negative,
suggesting that the seat would pull on the person, which cannot happen. If the
speed is less than vc the person cannot turn the full loop but will instead fall from
the car as a projectile. The same thing happens with water in a container being
rotated vertically. If the velocity is less than the critical velocity, the bucket will
not push against the water to provide the inward force that causes circular motion.
If the velocity is less than vc value the water leaves the bucket.
Example:
A motorcyclist in the Globe of Death rides in a 2.2 m radius vertical loop. To keep
control of the bike, the rider wants the normal force on the tires at the top of the loop to
equal or exceed the riders and the bike’s combined weight. What is the minimum speed
at which the rider can take the loop?
Centrifuges
Centrifuges are used to separate parts of a liquid that have different densities.
Small particles suspended in a liquid will settle out slowly. For biological
applications, such as blood analysis, the time required for this can be too long. A
centrifuge helps reduce the time for the settling out process. By spinning the liquid
very fast, the apparent weight of the particles is greatly increased, allowing them to
settle out quickly.
Example: An 18 cm diameter ultracentrifuge produces an extraordinarily large
centripetal acceleration of 250,000 g, where g is the free-fall
acceleration due to gravity. What is the frequency in rpm? What is the
apparent weight of a sample with a mass of 0.0030 kg?
Multiple Choice Example:
A car is rolling over the top of a hill at constant speed v. At this instant,
A. n > w
n
B. n < w
v
C. n = w
D. We cannot tell
about n without
knowing v.
w