Centripetal Acceleration
• Centripetal acceleration is the rate of change in
velocity of an object that is associated with the
change in direction of the velocity.
– Centripetal
acceleration is
always
perpendicular to
the velocity.
– Centripetal
acceleration always
points toward the
center of the curve.
v2
ac 
r
Centripetal Forces
• The centripetal force may be due to one or
more individual forces, such as a normal
force and/or a force due to friction.
.
2
v
ac 
r
Fc  mac
• The friction
between the
tires and road
produces the
centripetal
acceleration
on a level
curve.
 The normal force can be separated into a vertical
component and a ho
 The horizontal component of the normal force is
the centripetal force when there’s no friction in
order to keep the object (car) on the circular
track.
Quiz: A Car moves along a banked circular track
(θ=30o) and experience a normal force of 1000 kg.
How large is the vertical component (Fv) and
horizontal component (Fh)?
A). Fv = 866kg, Fh = 500kg.
B). Fv = 500 kg, Fh = 866 kg.
C). Fv = 500 kg, Fh = 500 kg
D). Fv = 866kg, Fh = 134 kg.
Fh = N sin(30o)
Fv = N cos(30o)
1D-02 Conical Pendulum
Could you
find the
NET force?
Vertical component of T balance
the mg.
T sin(θ) = mv2/R
T cos(θ) = mg
v = sqrt( gR tan(θ) )
Period of the pendulum
τ= 2πR/v,
where R = L / sin(θ)
τ= 2π sqrt( Lcos(θ)/g )
•NET FORCE IS TOWARD THE CENTER OF THE CIRCULAR PATH
•Why R become smaller as time goes by?
•T sin(θ) has to be smaller while T=mg/cos(θ) increase
2/4/2011
Physics 214 Fall 2010
with larger θ
6
Vertical circles
N
v
g
W = mg
mg – N = mv2/r
•If v = 0 then N = mg
+ is always
toward the
center of
the circle
At the bottom
N - mg = mv2/r
•As v increases N becomes smaller
• When v2/r = g the car becomes
weightless.
2/4/2011
Ferris wheel
At the top
Mg – N = mv2/r
Physics 214 Fall 2010
7
1D-05 Twirling Wine Glass
m
v
Is it possible .to
keep the water in
the cut upside
down?
g
Same as
N + mg = mv2/R
string
N>0
What ‘s the minimum speed to keep the water from spill?
• When N = 0,
• Vmin=
• If V < Vmin, it can not reach the top.
2/4/2011
Physics 214 Fall 2010
8
Ch 5 CP 2
A Ferris wheel with radius 12 m makes one complete
rotation every 8 seconds.
What speed do riders move at?
A).
B).
C).
D).
56.52 m/s
9.42 m/s
18.84 m/s
4.71 m/s
Fcent
S = d/t = 2r/t = 2(12m)/8s = 9.42 m/s
2/4/2011
Physics 214 Fall 2010
9
Ch 5 CP 2
A Ferris wheel with radius 12 m makes one complete
rotation every 8 seconds.
What is the magnitude of their centripetal acceleration?
A). 7.40 m/s2
B). 9.40 m/s2
C). 3.70 m/s2
D). 14.80 m/s2
Fcent
acent = v2/r = s2/r = (9.42m/s)2/12m = 7.40 m/s2
2/4/2011
Physics 214 Fall 2010
10
Ch 5 CP 2
A Ferris wheel with radius 12 m makes one complete
rotation every 8 seconds.
For a 40 kg rider, what is magnitude of centripetal force to keep him
moving in a circle? Is his weight large enough to provide this
centripetal force at the top of the cycle?
A). 396 N, weight is large enough.
B). 1028 N, weight is large enough
C). 200 N, weight is large enough
D). 296 N, weight is large enough
E). 296 N, weight is NOT large enough
Fcent
Fcent = m v2/r = m acent = (40 kg)(7.40 m/s2) = 296 N
W = mg = (40 kg)(9.8 m/s2) = 392 N
Yes, his weight is larger than the centripetal force required.
2/4/2011
Physics 214 Fall 2010
11
Ch 5 CP 2
A Ferris wheel with radius 12 m makes one complete
rotation every 8 seconds.
What is the magnitude of the normal force exerted by the seat on the
rider at the top?
A). 100 N
B). 96 N
C). 90 N
D). 50 N
E). 48 N
Fcent
W – Nf = 296
2/4/2011
N = 96 newtons
Physics 214 Fall 2010
12
Ch 5 CP 2
A Ferris wheel with radius 12 m makes one complete
rotation every 8 seconds.
What would happen if the Ferris wheel is going so fast the weight of the
rider is not sufficient to provide the centripetal force at the top? Note:
there’s no safe belt.
A) rider is ejected
B) Rider remain on seat
2/4/2011
Fcent
13
Planetary Motion
• The ancient Greeks believed the sun, moon,
stars and planets all revolved around the
Earth.
– This is called a geocentric view (Earth-centered) of
the universe.
– This view matched their observations of the sky,
with the exception of the puzzling motion of the
wandering planets.
Kepler’s First Law of Planetary Motion
Tycho’s assistant,
Kepler, analyzed all
that careful data.
Kepler was able to
show that the orbits of
the planets around the
sun are ellipses, with
the sun at one focus.
This is Kepler’s first
law of planetary motion.
Kepler’s Second Law of Planetary Motion
planets move faster
when nearer to the
sun, the radius line for
each planet sweeps
out equal areas in
equal times.
The two blue
sections each cover
the same span of time
and have equal area.
Kepler’s Third Law of Planetary Motion
2
T r
3
The period (T) of
an orbit is the time it
takes for one
complete cycle
around the sun.
The cube of the
average radius (r)
about the sun is
proportional to the
square of the period
of the orbit.
Newton’s Law of Universal Gravitation
http://www.physics.purdue.edu/class/applets/NewtonsCannon/ne
wtmtn.html
• Newton recognized the
similarity between the
motion of a projectile on
Earth and the orbit of the
moon.
• If a projectile is fired with
enough velocity, it could fall
towards Earth but never
reach the surface.
Newton’s Law of Universal Gravitation
• Newton was able to explain Kepler’s 1st and 3rd laws by
assuming the gravitational force between planets and
the sun falls off as the inverse square of the distance.
• Newton’s law of universal gravitation says the
gravitational force between two objects is proportional
to the mass of each object, and inversely proportional to
the square of the distance between the two objects.
•G is the Universal gravitational constant G, which was
measured by Cavendish after more than 100 years
•
G = 6.67 x 10-11 N.m2/kg2.
Gm1m2
F
r2
The gravitational force is attractive and acts along the
line joining the center of the two masses.
It obeys Newton’s third law of motion.
Gm1m2
F
r2
Using Newton’s Gravitational Law to
derive Kepler’s 3rd Law
•For a simple circular orbit
GmMs/r2 = mv2/r
•
•where M is the mass of the sun and
m the mass of the earth or M is the
mass of the earth and m the mass of
a satellite.
•Circumference of a circle:
•Period T = s/v =
2/4/2011
=
T2/r3 = 4π2/GMs
/v
Physics 214 Fall 2010
21
Quiz: Three equal masses are located
as shown. What is the direction of the
total force acting on m2?
a)
b)
c)
d)
To the left.
To the right.
The forces cancel such that the total force is zero.
It is impossible to determine from the figure.
There will be a net force acting on m2 toward m1. The third mass exerts a force of
attraction to the right, but since it is farther away that force is less than the force
exerted by m1 to the left.
Ch 5 E 14
The acceleration of gravity at the surface of the moon is
about 1/6 that at the surface of the Earth (9.8 m/s2). What
is the weight of an astronaut standing on the moon
whose weight on earth is 180 lb?
A). 30 lb
B). 180 lb.
C). 15 lb.
D). 200 lb
E). 215 lb.
2/4/2011
Physics 214 Fall 2010
23
Ch 5 E 14
The acceleration of gravity at the surface of the moon is
about 1/6 that at the surface of the Earth (9.8 m/s2). What
is the weight of an astronaut standing on the moon
whose weight on earth is 180 lb?
Wearth = m gearth = 180 lb
gmoo
Wmoon = m gmoon
n
gmoon = 1/6 gearth
Wmoon = m 1/6 gearth = 1/6 m gearth = 1/6 (180 lb)
2/4/2011
Physics 214 Fall 2010
24
Moon and tides
anim0012.mov
•Tides are dominantly due to the
gravitational force exerted by the moon.
Since the earth and moon are rotating this
effect also plays a role. The moon is locked
to the earth so that we always see the
same face. Because of the friction
generated by tides the moon is losing
energy and moving away from the earth.
http://www.sfgate.com/getoutside/1996/jun/tides.html
2/4/2011
Physics 214 Fall 2010
25
Ch 5 E 16
Time between high tides = 12 hrs 25 minutes.
High tide occurs at 3:30 PM one afternoon.
When is high tide the next afternoon?
a)
b)
c)
d)
e)
4:20 pm
5:20 pm
6.00 pm
3:00 pm
2:20 pm
2/4/2011
Physics 214 Fall 2010
26
Ch 5 E 16
Time between high tides = 12 hrs 25 minutes.
High tide occurs at 3:30 PM one afternoon.
When is high tide the next afternoon?
3:30 PM + 2 (12 hrs 25 min)
= 3:30 PM + 24 hrs + 50 min
= 4:20 PM
2/4/2011
Physics 214 Fall 2010
27
Ch 5 CP 4
A passenger in a rollover accident turns through a radius of
3.0m in the seat of the vehicle making a complete turn in 1
sec.
what is speed of passenger?
A). 38 m/s
B). 22 m/s
C). 19 m/s
D). 11 m/s
E). 125.3 m/s
3m
s = d/t = 2(3.0m)/1 = 19m/s
2/4/2011
Physics 214 Fall 2010
28
Quiz
A passenger in a rollover accident turns through a radius of
3.0m in the seat of the vehicle making a complete turn in 1
sec.
What is centripetal acceleration? Compare it to gravity (g = 9.8
m/s2)
A). 118 m/s2
B). 128 m/s2
C). 138 m/s2
D). 108 m/s2
E). 98 m/s2
a=
v2/r
2/4/2011
=
s2/r
= (19
m/s)2/3m
= 118
m/s2
= 12xg
Physics 214 Fall 2010
3m
29
Ch 5 CP 4
A passenger in a rollover accident turns through a radius of
3.0m in the seat of the vehicle making a complete turn in 1
sec.
Passenger has mass = 60 kg, what is centripetal force required
to produce the acceleration? Compare it to passengers weight.
A). 10000 N
B). 1352 N
C). 7080 N
D). 1159.3 N
E). 205 N
m/s2)
F = ma = (60 kg)(118
= 7080 N
F = ma = m (12 X 9.8m/s2) = 12 mg = 12
weight
2/4/2011
Physics 214 Fall 2010
3m
30
Ch 5 E 16
Time between high tides = 12 hrs 25 minutes.
High tide occurs at 3:30 PM one afternoon.
When are the 1st and 2nd low tides the next day?
a). First one is at 10:07:30 AM, 2nd one is at 10:32:30 PM
b). First one is at 8:07:30 AM, 2nd one is at 8:32:30 PM
c). First one is at 6:07:30 AM, 2nd one is at 6:32:30 PM
d). First one is at 5:07:30 AM, 2nd one is at 5:32:30 PM
e). First one is at 11:07:30 AM, 2nd one is at 11:32:30 PM
2/4/2011
Physics 214 Fall 2010
31
Ch 5 E 16
Time between high tides = 12 hrs 25 minutes.
High tide occurs at 3:30 PM one afternoon.
When are the 1st and 2nd low tides the next
day?
High tide next afternoon:
3:30 PM + 2 (12 hrs 25 min)
= 3:30 PM + 24 hrs + 50 min
= 4:20 PM
b) Low tide the next day = 4:20 PM - 6 hr 12 min 30 s = 10:07:30 AM
2nd Low tide = 10:07:30 AM + 12 hrs 25 min = 10:32:30 PM
2/4/2011
Physics 214 Fall 2010
32
How Tough is 12 g?
What’s shown in the video is < 10g.