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Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Lecture 12
Chapter 8
Uniform Circular MotionGravity
Physics I
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Today we are going to discuss:
Chapter 8:
Uniform Circular Motion: Section 8.2 Circular Orbits: Section 8.3 Reasoning about Circular Motion: Section 8.4
IN THIS CHAPTER, you will learn to solve problems about motion in two dimensions.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Examples. Banked curve
But sometimes, friction force is not enough to keep a car on a circular road.
Banking the curve can help to keep cars from skidding.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
mg
Ncos
Nsin
Fz = Ncos mg = 0
Fr = Nsin = marar= v2/R
N
Nsin mv2/R
tan v2/gRIndependent of object mass !!!
Banked Curves (solution)
r component of normal force provides the centripetal accelerationhttp://phys23p.sl.psu.edu/phys_anim/mech/car_banked_new.avi
r
z
Ncos mg
z
r
Take a ratio
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Example: Loop the Loop
http://phys23p.sl.psu.edu/phys_anim/mech/car_vert_bare.avi
http://phys23p.sl.psu.edu/phys_anim/mech/car_vert_fail.avi
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Loop the LoopTo make the loop-the-loop at a constant speed, what minimum speed does the motorcycle need?
RvmmgN
2
gm
v
R
v
N
gm
N
1
2
r
r
2 rr maF Rvar
2
mgNmRv
When N=0 (feels like no weight), then speed is minimum
gRv http://phys23p.sl.psu.edu/phys_anim/mech/car_vert_bare.avi
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Loop the LoopApparent weight at the bottom, point 1?
RvmmgN
2
R
v
gm
N
1
r
1 rr maF Rvar
2
Thus, N > mg. You would feel heavier (similar to an elevator)
mgRvmN
2
http://phys23p.sl.psu.edu/phys_anim/mech/car_vert_bare.avi
ConcepTest Going in Circles
A) N remains equal to mg
B) N is smaller than mg
C) N is larger than mg
D) none of the above
You’re on a Ferris wheel moving in a vertical circle. When the Ferris wheel is at rest, the normal force N exerted by your seat is equal to your weight mg. How does N change at the top of the Ferris wheel when you are in motion?
You are in circular motion, so there has to be a
centripetal force pointing inward. At the top, the
only two forces are mg (down) and N (up), so N
must be smaller than mg.
Follow-up: Where is N larger than mg?
RvmNmg
2
NRvmmg
2
N
RvmmgN
2
RvmmgN
2
Bottom
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
The motion of Satellites
Section 6.3Section 8.3
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
universal gravitational constant
M
m
F
Gravitational force is central, attractive, proportional to masses, and inversely proportional to the square of the distance
2rmMGF
Newton’s Law of Universal Gravitation
G 6.671011 N m2 kg2
A gravitational force of attraction exists between all objects that have mass
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Gravity near the Earth’s SurfaceWhat should we get for the acceleration due to gravity at the Earth’s surface?
MEARTH 5.981024 kg REARTH 6.38106 m G 6.671011 Nm2
kg2
mg
g GME
RE2 9.80 m/s2
2rmGMF E
2E
E
RmGMF
ME
m Let’s apply N.2nd law for gravitational force
maF
2rmGMma E
Denote the acceleration due to gravity as 2)(
rGMrga E
If r=RE (on the surface)
F
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Newton’s Cannon on a Mountain
http://waowen.screaming.net/revision/force&motion/ncananim.htm
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Example:
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Weighlessness (Explanation)
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
The “weightlessness” in orbit is exactly the same, everything is accelerating equally independent of mass.
The satellite and all its contents are in free fall, except with a huge tangential velocity. So, there is no normal force. This is what leads to the experience of weightlessness.
“Weightlessness”
They do have a gravitational force acting on them, though!
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Thank youSee you on Wednesday
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
mg
Tz
Tr
ar
Example: Conical pendulum
T
z
r
)1(0 mgTF zz
)2(2
RmvSinTTr
rrr maTF
r component of tension provides the centripetal acceleration
R
vFc points toward the center of the circle (i.e., downward in this case). The weight vectorpoints down and the normal force (exerted by the hill) points up. The magnitude of the net force, therefore, isFc = mg – N.
mg N
A skier goes over a small round hill with radius R. Because she is in circular motion, there has to be a centripetal force. At the top of the hill, what is Fc of the skier equal to?
ConcepTest 4 Going in Circles II
Follow-up: What happens when the skier goes into a small dip?
A) Fc = N + mg
B) Fc = mg – N
C) Fc = T + N – mg
D) Fc = N
E) Fc = mg
Department of Physics and Applied PhysicsPHYS.1410 Lecture 12 Danylov
Variations in “g” on the surface
The acceleration due to gravityvaries over the Earth’s surface dueto altitude, local geology, and theshape of the Earth, which is notquite spherical.