CIRCULAR MOTION

Path of an object in circular motion:

• The velocity is tangential

• The acceleration is directed towards the center (centripetal acceleration)

Centripetal – center seeking

Centripetal Acceleration

• Velocity’s direction is changing BUT magnitude remains the same

ac = v2

r

ac = centripetal acceleration

v2 = velocity

r = radius

***This acceleration is always directed toward the center***

Centripetal Acceleration

• ac = v2 v= 2r

r T

• Where v= velocity, r= radius, T= period

• ac = (2r/T)2 / r ac = 42r / T2

Centripetal Force

• F=ma a = centripetal acceleration

• F c = mv2

r

F c = m42r / T2

example problem : An object weighing 49N moves in a circular path of radius 0.5 meter at a speed of 10m/s. Calculate the centripetal acceleration and the centripetal force.

Vertical Circles

• When objects (planes) move in vertical circles, there are two forces acting on the body (within)

• Recall that F c = net force

Vertical CirclesA person sitting in a plane experiences Wt and the force exerted by the seat on

him/her. These forces together result in the centripetal force:

Fnet = Fc = Fseat + mg

when the plane is upside down:

(-Fc) = (-Fseat)+ (-mg) (-mv2 )= (-Fseat) + (-mg) r

Fseat = (mv2 )- mg seat requires less force!

r

Vertical Circles

When the plane is upright:

Fc = Fseat + (-Wt)

mv2 = Fseat - mg

r

Fseat = mv2 + mg seat requires more

force! r

g -Forces

• In vertical problems, centripetal acceleration adds to the gravitational acceleration…

• At the top of the loop:

gravity and centripetal acceleration act in the same direction

Looping Roller Coasters

How do the cars stay on the track?

~the forward motion (velocity) and the curve of the track

The forward velocity value must be large enough…because of inertia, it will want to continue in a straight line…the car will push on the track, keeping it secure.

Looping Roller Coasters (Cont.)

• Newton’s 3rd Law- the tracks exert a force on the car and the car exerts a force back on the track

The curve or radius determines the acceleration…

a = v2/r a and r are inversely related

The smaller the curve, the greater the acceleration..

The greater the acceleration the greater the centripetal force.

Centrifugal Force

• Centrifugal force – non-existent force…

• It is really the effect of the body’s inertia

• The body wants to move forward in a straight line, but the centripetal acceleration pulls it in a curved path. The inertia feels like a force pulling outward.

Universal Gravitation

Isaac Newton (45 years after Kepler’s work) hypothesized that forces on bodies are proportional to their masses, and that according to his third law, bodies exert forces on each other equal and opposite. As gravity pulls on us, so it pulls on other planets, the sun, etc..

Inverse-Square Law

F α 1 d2

The force on a body (planet) is inversely proportional to the square of the distance.

• Newton took Kepler’s work with ellipses and mathematically came up with this law. He said that forces can be applied in the heavens as they do on Earth.

Gravitational force- force of attraction that exists b/t all bodies

Law of Universal Gravitation

F = G ma mb d2

G=Gravitational constant = 6.67 x 10-11 N m2/kg2

d = distance between centers of the two bodies

Satellite Motion

• Simplifying gravitation…assume a circular orbit Fc = Fg

• mAac = GmAmΘ A = planet Θ = Earth

r2

• mA4Π2r = GmAmΘ

T2 r2

• T2 = 4Π2 r3

GmΘ

Satellite Motion

• All launched objects take a projectile path (trajectory) • Newton predicted that an object, if it has enough speed

will fall according to gravity, but maintain the same height above the Earth.

• It is continually falling towards the Earth, but the curvature of the Earth is such that the distance b/t the Earth and the satellite remains the same

• If the satellite loses speed, it would fall to Earth• If it increases speed enough, it would escape its orbit

Velocity of Satellites

Fc = Fg

msv2 = GmsmΘ s = satellite Θ = Earth

r r2

v = G mΘ

r***Both the velocity and period of a satellite are

independent of its mass***

Geosynchronous orbit – satellite above the Earth whose period of revolution is equal to one rotation of the Earth on its axis (24hrs). The satellite remains fixed on one spot over the Earth (equatorial region)