6. Centripetal force F = ma 1. Example: A stone of mass m sits at the bottom of a bucket. A string is attached to the bucket and the whole thing is made.

6. Centripetal force

F = ma rT

mrmr

mvF

22

2 2

1

Example: A stone of mass m sits at the bottom of a bucket. A string is attached to the bucket and the whole thing is made to move in circles. What is the minimum speed that the bucket needs to have at the highest point of the trajectory in order to keep the stone inside the bucket?

mgr

mvN

r

mvmgN

r

mvF

222

grv min0min N

2

Examples (centripetal force)

(3)

(2)

(1) mgN

r

mvmgN

2

r

mvmgN

2

mgr

mvN

2

mgr

mvN

2

r

mvmgN

2

r

mvNmg

2

mg

N

N

mg

F = mar

mvF

2

3

Question 1: The ball whirls around a pole.In what direction does the net force on the ball point?1) toward the top of the pole2) toward the ground3) along the horizontal component of the tension force4) along the vertical component of the tension force5) tangential to the circle

mg

T

Question 2: A particle moves at a constant speed along the presented trajectory. Compare the magnitude of the acceleration of the particle at points A and B.

A

B

The vertical component of the tension balances the weight. The horizontal component of tension provides the centripetal force that points toward the center of the circle.

Question 3: A ball is going around in a circle at constant speed. What is the angle between the acceleration vector and the velocity vector of the ball?

D)180 90 C) 45 B) 0 A)

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Newton’s Law of Gravitation

221

r

mGmFg

2211 /1067.6 kgmNG

Gravitational force:• one of the fundamental forces of nature• always attractive• exist between any two objects and always act along the line joining

the two objects• one of the fundamental forces acting in our galaxy and the main force

of interaction between the sun and planets including Earth

2m

12 onF

r1m

5

Gravitational constant:

21onF

For determining of G, Henry Cavendish in 1798 used an instrument

called a torsion balance. A modern version of the Cavendish torsion

balance is shown below.

1. Determining the value of G

21

2

mm

rFG g

6

The weight of a body is the total gravitational force acting on that body

Consider an object near

the surface of the earth:

mgr

MmGFg

2

2. Weight and Law of Gravitation

What happened if

the object will move

far from the earth?

2r

MGg

7

gF

Question 1: The mass of the earth is 81 times the mass of the moon. The magnitude of the gravitational force of Earth on the Moon is __ times the magnitude of the gravitational force of the Moon on the Earth. A. 81 B. 812 C. 1 D. 1/81 E. (1/81)2

1. up 2. down 3. to the right4. to the left5. There is no

net force on m

Question 2: The direction of the net gravitational force on m due to the two masses M is ___.

Question 3: The planet Saturn has 100 times the mass of Earth. Saturn is 10 times further from the Sun than Earth is. The magnitude of the acceleration of Earth in its orbit around the Sun is ___times the acceleration of Saturn.A. 100 B. 10 C. 1 D. 1/10 E. 1/100

?

0123.0

273.0

EarthMoon

EarthMoon

EarthMoon

gg

MM

RRQuestion 4: A. 0.06 B. 0.17C. 0.39D. 0.62 8

3. Orbits of planets and satellites(Fundamentally important application of Newton’s Mechanics)

•Copernicus in 1543 proposed that the sun was the center of the Solar System with the planets moving in circular orbits.

•In 1619 Kepler showed that planets followed elliptical orbits using huge amount of high quality data gathered by Tycho Brahe by naked eye astronomy. •Kepler characterized planetary orbits using “Kepler’s Three Laws”.

•In 1683 Newton showed that Kepler’s laws follow from his “Law of Gravity” and his “Three Laws of Mechanics”.

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3a. Circular Orbits

2/1

2/3

)(

22

EGm

r

v

rT

Velocity:

Period:

r

mv

r

mmG E

2

2

r

Gmv E

Example: We want to place 2000-kg satellite into a circular orbit 400 km

above the earth’s surface. For the earth: RE = 6380 km, mE = 5.971024kg.

min6.925556/7664

)1078.6(22 6

ssm

m

v

rT

smm

kgkgmN

r

Gmv E /7664

)104.01038.6(

)1097.5)(/1067.6(66

242211

10

3b. Satellites Orbits

rGmv Ecircular /If r = RE = 6380 km then

7) hyperbolic orbit: v > vescape

rGmv Eescape /2

6) parabolic orbit: v = vescape

5) elliptical orbit: vescape>v >vcircular

4) circular orbit: v = vcircular

1,2,3) elliptical orbit: v < vcircular

km/s 7.9circularv

km/s 11.2escapev

Trajectories of a projectile launched from point A in the direction AB with different speeds:

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Question: The Moon does not crash into Earth because:1) it’s in Earth’s gravitational field2) the net force on it is zero3) it is beyond the main pull of Earth’s gravity4) it’s being pulled by the Sun as well as by Earth5) none of the above

The Moon does not crash into Earth because of its high speed. If it stopped moving, it would, of course, fall directly into Earth. With its high speed, the Moon would fly off into space if it weren’t for gravity providing the centripetal force.

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