Circular Motion
There must be a net inward force: Centripetal Force
v2
aC = R
FC = maC
Centripetal acceleration
R
evolutions
P M
er
v
v
2πR
v = T
R
v
inute
v
(n)2πR
v = 60 s
An Ipod's hard drive rotates at 4200 RPM. Its radius is 0.03 m. What is the velocity in m/s of its edge? FC
Solving Circular Motion Problems
1. Determine FC (FBD)
2. Set FC = mv2
R
3. Solve for unknowns
The book's mass is 2 kg it is spun so that the tension is 45 N. What is the velocity of the book in m/s? In RPM?
R = 0.75 m
[4.11 m/s]
[52 RPM]
What is the period (T) of the book?
[1.15 s]
The car is traveling at 25 m/s and the turn's radius is
63 m. What must the μ be so it will not slip?
[1.01]
What must be the velocity of the ride if μ = 0.85?
R = 2.2 m
[5.04 m/s]
Other circles
What is the velocity?
13o
0.85 m
2 kg
[0.66 m/s]
What is the period?
[1.83 s]
Banked turns (frictionless road)
R = 220 m, v = 32 m/s. What must be the θ?
N
Ncosθ
θ
Nsinθ
mg
θ
[25.4o]
What is the turn radius of the plane (v = 53 m/s)?
10o
R = ?
Vertical Circles
If the car maintains a constant speed, what is the normal force at the top of the loop (m = 2.5 kg) ?
5 m
v = 14 m/s
[73.5 N]
What is the minimum speed needed for the car to make the loop without falling?
[7 m/s]
What is the maximum speed the car can go and not lose contact with the road at A? If it maintains that speed, what will be its normal force at B?
m = 750 kg
R = 100 m
R = 100 m
[vA = 31.3 m/s]
[NB = 14,700 N]
Gravity
•
•
•
•
ALWAYS attracts
More mass = more gravity
Further away = less gravity
Weak, but acts over long distances
FG
FG
m2
R
m1
FG = G m1 m2
R2
G = 6.67x10­11N
m2
kg2
3.05 m (10 ft)
14,000 kg
5,000 kg
F = 1.99x1020 N
Earth
ME = 5.98x1024 kg
1
[0.0005 N about th a paperclip]
20
R = ? m
Moon
MM = 7.35x1022 kg
[3.84x108 m]
[3.84E8 m]
What is the force on the 1.5x108 kg mass?
8x108 kg
0.75 m
3x108 kg
1.5x108 kg
1.15 m
g Anywhere....
Not to scale!
Venus
RV = 6.05x106 m
MV = 4.87x1024 kg
[gJ = 8.87 m/s2]
How far from the surface of the Earth is the value of g
1
less than at the surface?
3
?
RE=6.38x106 m
[h = 4.67x106 m]
Einstein
Mass warps space­time
Warped space­time = gravity
Satellite Motion
R = 8.2x106 m
ME = 5.98x1024 kg
What is the velocity of the satellite?
[6976 m/s]
What is the period of the satellite?
[7385 s]
[123 min]
[2.1 hr]
GEOSTAT
GPS
LOW
ISS
The rocket is in orbit 215 km above the Earth. What must its period be?
215 km
RE = 6.38x106 m
T = ?
ME = 5.98x1024 kg
[5328 s]
Kepler's Laws
1. All planets move in elliptical orbits with the Sun at one focus.
2. A line drawn from the Sun to any planet sweeps out equal areas in equal
time intervals.
3. Equation relating 2 orbiting bodies.
(
(
(
2
T1
=
T2
(
1.5x1011 m = 1A.U.
RX = 3RY. How many times longer is the period of X?
If X travels 90o, how far will Y travel?
R1
R2
3
Summary
FG = G m1 m2
2
R
G m1 m2
m1g =
R2
m1 v2 G m1 m2
=
R
R2
(Forces)
("g")
(one orbit)
may use v = 2πR
T
(2 orbits)
R IS ALWAYS CENTER TO CENTER!!
Attachments
cavendish1.avi