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http://www.physicsclassroom.com/mmedia/circmot/circmotTOC.html

Uniform Circular Motion

motion of an object in a circle with a constant or uniform speed

constant change in direction

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The direction of the velocity vector at everyinstant is in the direction tangent to the circle

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Uniform Circular Motion: Period

What is this uniform speed?

distance = rate time

time =distance

rate v

T =2 r

v

2

r

The speed of the object is the distance it covers ( the circumference of the circle, 2лr) divided by time T (period or the to make one complete revolution

Are all people on Earth moving at the same speed?

Earth is rotating about an axis through its poles

So that means we are all moving since we are all on the Earth.

Are some of us moving with a greater LINEAR SPEED than others?? Yes, closer to the Equator, the

faster you are moving…. Closer to poles, the slower you are moving

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The speed is constant but the direction is changing with time

A

r

dh

h = dv

r2

vh

Equation 1dh = vht

Equation 2dv = ½act²

Equation 3dh² = 2rdv

CENTRIPETAL ACCELERATION

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Centripetal Acceleration

Substitute Eq. 1 & Eq. 2into Eq. 3

Equation 4. ac = vh² rSince vh = 2лr/T, then

Equation 5. ac = 4л²r T²If T= 1/f, then

Equation 6. ac = 4л²rf²

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CENTRIPETAL FORCE

F ma

Fmv

r

2 Centripetal Force

Centripetal force is the magnitude of the force required to maintain uniform circular motion.

Always points toward center of circle. (Always changing direction!)

Centripetal means “center- seeking”. The force pushes you toward the center of the circle and keeps you moving in a circle. It keeps your inertia from taking you in a straight line. Centripetal Force is affected by mass (m), linear speed(vt), and radius (r)

Direction of Centripetal Force, Acceleration and Velocity

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Without a centripetal force, an object in motion continues along a

straight-line path.

Direction of Centripetal Force, Acceleration and Velocity

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1. When driving in a circle, in what direction is a force acting on you? Pushing you outward from the circle, or inward?

Ans. -- Inwards, toward the center of the circle

2. If you are swinging a yo-yo in a circle, and the string breaks…. What path does the yo – yo take?

Ans -- yo- yo goes in a path tangent to the circle

HOWEVER, People commonly think there is a force pushing you out from the circle; Feels like you are being pushed outward

Example : The Rotor- amusement park ride, a centrifuge, CD on your dashboard moving to the right when your turning left . Why is this?

What if velocity decreases?

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What if mass decreases?

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What if radius decreases?

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What provides the centripetal force?

Tension GravityFriction Normal Force

Centripetal force is NOT a fundamental force. Acceleration is the result of the net force and

centripetal force is derived only from the Newton’s 2nd law. So centripetal force depends only on the net force applied to a body moving

in uniform circular motion

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Tension Can Yield a Centripetal Acceleration

If the person doubles the speed of the airplane, what happens to the tension in the cable?

F = mamv

r

2

Doubling the speed, quadruples the force (tension) required to keep the plane in uniform circular motion.

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Friction Can Yield a Centripetal Acceleration:

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Frictional force between the tire and the road provides the centripetal acceleration. For the car to make the turn without skidding, a minimum

coefficient of static friction must be present between the tire and the road

Example: A car traveling around a circular track

Fc = Ffmv²/r = µFn

µ = v² rg

Sample problem

A car is traveling at 9 m/s in a circle that has a radius of 60 m. What must be the minimum value of µ for the car to make turn without skidding?

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Given: v= 9m/s r= 60 m

Find: µ µ = v²/rg

Solution: µ = v²/rg = (9m/s)²/ (60m)(9.8 m/s²)

Answer: µ = 0.14

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Gravity Can Yield a Centripetal Acceleration

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Gravity and Centripetal Acceleration

Centripetal acceleration provided by gravitational force

G m M

R

m v

RE

2

2

vG m M R

m R

vG M

R

vG M

R

E

E

E

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2

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Hubble Space Telescope orbits at an altitude of 598 km (height above Earth’s surface).What is its orbital speed?

vGM

R km

v

v

E

E

598

6 67 10 974 10

7 600

11 24( . ) (5.

,

m kg s kg)

6,976,000 m

m / s

3 -1 -2

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The Normal Force Can Yield a Centripetal Acceleration:

Engineers have learned to “bank” curves so that cars can safely travel around the curve without relying on friction at all to supply the centripetal acceleration.

Banked Curves

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Q: Why exit ramps in highways are banked?

Answer: To increase the centripetal force for the higher exit speed.

Banked Curves

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FN cosq = mg Fc = FN sinq = mv²

r

r

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Fw=FN cosq

mg = FN cosq

Fc =FN sinq

mv²/r = FN sinq

Fc = mv²/r = FN sinq

Fw mg FN cosq

v²/rg = tan q = q tan-1 v²/rg

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The Normal Force and Centripetal Acceleration:

tan v

gr

2

How to bank a curve…

…so that you don’t rely on friction at all

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Given: v= 27 m/s r= 60 m

Find : ƟƟ = tan-1 v²/rg

Solution: Ɵ = tan-1 v²/rg Ɵ = tan-1 (27m/s) ² (60 m) (9.8m/s²) Ɵ = 51º

Sample Problem: A car is to make turn with a radiusof curvature of 60 m at a speed of27 m/s. at what angle should the road be banked for the car to make the turn?

Vertical Circular Motion

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Artifical Gravity

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A rotating room has a radius of 4.5 m and the speed of the rider is 12 m/s. how much should the coefficient of static friction be to keep the rider pinned against the wall?

Given: r = 4.5 m v= 12/ms

Find µ µ = rg/v²

Solution: µ = rg/v² µ = (4.5m)(9.8 m/s²)/ (12 m/s) ²µ = 0.31

Sample Problem

The RotorPeople stand with backs against wall of a large cylinder, cylinder then starts spinning, and people are seemingly pushed against the wall, then floor drops, and people are stuck against the wall.

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A force does not cause this…… your INERTIA does! Inertia makes you want to stay in a straight line, and by going in a circle, you are fighting your own inertia

Ex: This is how Rotor works, and why CD on dashboard happens, The only actual force acting on you is the

Centripetal Force

So why is there no Force pushing you out from the circle?

“Centrifugal Force“centrifugal force” is a fictitious

force - it is not an interaction between 2 objects, and therefore not a real force.Nothing pulls an object away from the center of the circle.

“Centrifugal Force”What is erroneously

attributed to “centrifugal force” is actually the action

of the object’s inertia - whatever velocity it has

(speed + direction) it wants to keep.

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Relationship Between Variables of Uniform Circular Motion

Suppose two identical objects go around in horizontal circles of identical diameter but one object goes around the circle twice as fast as the other. The force required to keep the faster object on the circular path is

A. the same as

B. one fourth of

C. half of

D. twice

E. four times

the force required to keep the slower object on the path.

The answer is E. As the velocity increases the centripetal force required to maintain the circle increases as the square of the speed.

Relationship Between Variables of Uniform Circular Motion

Suppose two identical objects go around in horizontal circles with the same speed. The diameter of one circle is half of the diameter of the other. The force required to keep the object on the smaller circular path is the same as

A. one fourth of B. half of C. twice D. four times

the force required to keep the object on the larger path.

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The answer is D. The centripetal force needed to maintain the circular motion of an object is inversely proportional to the radius of the circle. Everybody knows that it is harder to navigate a sharp turn than a wide turn.

Relationship Between Variables of Uniform Circular Motion

Suppose two identical objects go around in horizontal circles of identical diameter and speed but one object has twice the mass of the other. The force required to keep the more massive object on the circular path is

A. the same as

B. one fourth of

C. half of

D. twice

E. four times 41

Answer: D.The mass is directly proportional to centripetal force.

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F = mamv

r

2

Centripetal Force: Question

Smaller radius: larger force required to keep it in uniform circular motion.

A car travels at a constant speed around two curves. Where is the car most likely to skid? Why?

The End!

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