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.

rvac

2

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.

.

ac v 2

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)

2/4/2011 Physics 214 Fall 2010 6

1D-02 Conical Pendulum

•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 with larger θ

Vertical component of T balancethe mg.

T sin(θ) = mv2/RT cos(θ) = mgv = sqrt( gR tan(θ) )

Period of the pendulumτ= 2πR/v,where R = L / sin(θ)τ= 2π sqrt( Lcos(θ)/g )

Could you find the

NET force?

2/4/2011 Physics 214 Fall 2010 7

Vertical circles

•If v = 0 then N = mg

•As v increases N becomes smaller

• When v2/r = g the car becomes weightless.

mg – N = mv2/rFerris wheel

g

At the bottomN - mg = mv2/r

At the top Mg – N = mv2/r

+ is always toward the center of the circle

W = mg

N

v

2/4/2011 Physics 214 Fall 2010 8

1D-05 Twirling Wine Glass

g

Same as

mv

string

Is it possible .to keep the water in the cut upside down?

N + mg = mv2/R 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 9

A Ferris wheel with radius 12 m makes one complete rotation every 8 seconds.What speed do riders move at?

Ch 5 CP 2

S = d/t = 2r/t = 2(12m)/8s = 9.42 m/s

Fcent

A). 56.52 m/sB). 9.42 m/sC). 18.84 m/sD). 4.71 m/s

2/4/2011 Physics 214 Fall 2010 10

A Ferris wheel with radius 12 m makes one complete rotation every 8 seconds.What is the magnitude of their centripetal acceleration?

Ch 5 CP 2

Fcent

A). 7.40 m/s2

B). 9.40 m/s2

C). 3.70 m/s2

D). 14.80 m/s2

acent = v2/r = s2/r = (9.42m/s)2/12m = 7.40 m/s2

2/4/2011 Physics 214 Fall 2010 11

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?

Ch 5 CP 2

Fcent = m v2/r = m acent = (40 kg)(7.40 m/s2) = 296 NW = mg = (40 kg)(9.8 m/s2) = 392 NYes, his weight is larger than the centripetal force required.

Fcent

A). 396 N, weight is large enough. B). 1028 N, weight is large enoughC). 200 N, weight is large enoughD). 296 N, weight is large enoughE). 296 N, weight is NOT large enough

2/4/2011 Physics 214 Fall 2010 12

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?

Ch 5 CP 2

Fcent

W – Nf = 296 N = 96 newtons

A). 100 NB). 96 NC). 90 ND). 50 NE). 48 N

2/4/2011 13

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.

Ch 5 CP 2

A) rider is ejectedB) Rider remain on seat Fcent

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

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.

T 2 r3

Newton’s Law of Universal Gravitation

• 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.

http://www.physics.purdue.edu/class/applets/NewtonsCannon/newtmtn.html

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.

F Gm1m2

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.

F Gm1m2

r2

2/4/2011 Physics 214 Fall 2010 21

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 = /v

T2/r3 = 4π2/GMs

Quiz: Three equal masses are located as shown. What is the direction of the

total force acting on m2?

a) To the left.b) To the right.c) The forces cancel such that the total force is zero.d) 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.

2/4/2011 Physics 214 Fall 2010 23

A). 30 lb

B). 180 lb.

C). 15 lb.

D). 200 lb

E). 215 lb.

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?

Ch 5 E 14

2/4/2011 Physics 214 Fall 2010 24

Wearth = m gearth = 180 lb

Wmoon = m gmoon

gmoon = 1/6 gearth

Wmoon = m 1/6 gearth = 1/6 m gearth = 1/6 (180 lb)

gmoo

n

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?

Ch 5 E 14

2/4/2011 Physics 214 Fall 2010 25

Moon and tides•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

anim0012.mov

2/4/2011 Physics 214 Fall 2010 26

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) 4:20 pmb) 5:20 pmc) 6.00 pm d) 3:00 pm e) 2:20 pm

Ch 5 E 16

2/4/2011 Physics 214 Fall 2010 27

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

Ch 5 E 16

2/4/2011 Physics 214 Fall 2010 28

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?

3 ms = d/t = 2(3.0m)/1 = 19m/s

Ch 5 CP 4

A). 38 m/s B). 22 m/s C). 19 m/s D). 11 m/sE). 125.3 m/s

2/4/2011 Physics 214 Fall 2010 29

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)

3 ma = v2/r = s2/r = (19 m/s)2/3m = 118 m/s2 = 12xg

Quiz

A). 118 m/s2

B). 128 m/s2

C). 138 m/s2

D). 108 m/s2

E). 98 m/s2

2/4/2011 Physics 214 Fall 2010 30

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.

3 mF = ma = (60 kg)(118 m/s2) = 7080 NF = ma = m (12 X 9.8m/s2) = 12 mg = 12 weight

Ch 5 CP 4

A). 10000 N B). 1352 NC). 7080 N D). 1159.3 N E). 205 N

2/4/2011 Physics 214 Fall 2010 31

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 PMb). First one is at 8:07:30 AM, 2nd one is at 8:32:30 PMc). First one is at 6:07:30 AM, 2nd one is at 6:32:30 PMd). First one is at 5:07:30 AM, 2nd one is at 5:32:30 PMe). First one is at 11:07:30 AM, 2nd one is at 11:32:30 PM

Ch 5 E 16

2/4/2011 Physics 214 Fall 2010 32

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 AM2nd Low tide = 10:07:30 AM + 12 hrs 25 min = 10:32:30 PM

Ch 5 E 16

How Tough is 12 g?

What’s shown in the video is < 10g.