Circular Motion

When an object travels about a given point at a set distance it is

said to be in circular motion

Cause of Circular Motion

– 1st Law…an object in motion stays in

motion, in a straight line, at a constant speed unless acted on by an outside force.

– 2nd Law…an outside force causes an object to accelerate…a= F/m

– THEREFORE, circular motion is caused by a force that causes an object to travel contrary to its inertial path

Circular Motion Analysisv1

v2

rr

Circular Motion Analysisv1

v2

rr0

v1

v2

v = v2 - v1

or v = v2 + (-v1)

(-v1) = the opposite of v1

v1

(-v1)

v1

v2

rr0 v = v2 - v1

or v = v2 + (-v1)

(-v1) = the opposite of v1

v1

(-v1)

v1

v2

v2

(-v1)

vNote how v is directed toward the center of thecircle

v1

v2

rr

v1

v2

v2

(-v1)

v

l

Because the two triangles aresimilar, the angles are equal andthe ratio of the sides areproportional

v1

v2

rr

v1

v2

v2

(-v1)

v

l

Therefore,

v/v ~ l/r and v = vl/r

now, if a = v/t and v = vl/r

then, a = vl/rt since v = l/t

THEN, a = v2/r

Centripetal Acceleration

ac = v2/r

now, v = d/t and, d = c = 2r

then, v = 2r/t and, ac = (2r/t)2/r

or, ac = 42 r2/t2/r or, ac = 42r/T2

The 2nd Law and Centripetal Acceleration

Fc

ac

vt

F = ma

ac = v2/r = 42r/T2

therefore,

Fc = mv2/r or,

Fc = m42r/T2

Motion in a Vertical Circle

A

B

Fw TA

Fw

TB

Vertical circle

Fw TA

Fw

TB

A

B

Top of Circle

at vmin TA = 0 and Fw = Fc

therefore, TA + mg = mv2/r

because TA = 0, mg = mv2/r

and v2 = rg

Vertical Circle

Fw TA

Fw

TB

A

B

Bottom of Circle

vmax at bottom

thereore, TB - mg = mv2/r

or Fc = TB - Fw

or TB = mv2/r + mg

Cornering on the HorizontalWhen an object is caused to travel in a circular path because of the force of friction,then,...

Fc = FF

car

Fw

FN

FF

Cornering on the Horizontal

Fc = FF

car

Fw

FN

FF

Therefore, mv2/r = FN

because FN = Fw = mg,

mv2/r = mg … or,

= v2 /rg

Cornering on a Banked Curve

car

Fw FN

Fp

Cornering on a Banked Curve

car

Fw FN

Fc

Fw FN

Fc

Note how FN isthe Resultant

Fw FN

Fc

If we want to know the anglethe curve has to be at to allowthe car to circle without friction,then we have to analyze theforces acting on the car.

Sin = Fc/FN and, Fc = SinFN