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 –1 st Law…an object in motion stays in motion, in a straight line, at a constant speed unless acted on by an outside force. –2 nd 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 Analysis v1v1 v2v2 r r 

v1v1 v2v2 r r 0 v1v1 v2v2  v = v 2 - v 1 or  v = v 2 + (-v 1 ) (-v 1 ) = the opposite of v 1 v1v1 (-v 1 )

v1v1 v2v2 r r 0  v = v 2 - v 1 or  v = v 2 + (-v 1 ) (-v 1 ) = the opposite of v 1 v1v1 (-v 1 ) v1v1 v2v2 v2v2 vv Note how  v is directed toward the center of the circle

v1v1 v2v2 r r  v1v1 v2v2 v2v2 (-v 1 ) vv ll  Because the two triangles are similar, the angles are equal and the ratio of the sides are proportional

v1v1 v2v2 r r  v1v1 v2v2 v2v2 (-v 1 ) vv ll  Therefore,  v/v ~  l/rand  v = v  l/r now, if a =  v/tand  v = v  l/r then, a = v  l/rtsince v =  l/t THEN, a = v 2 /r

Centripetal Acceleration a c = v 2 /r now, v = d/t and, d = c = 2  r then, v = 2  r/t and, a c = (2  r/t) 2 /r or, a c = 4  2 r 2 /t 2 /r a c = 4  2 r/T 2

The 2 nd Law and Centripetal Acceleration FcFc acac vtvt F = ma a c = v 2 /r = 4  2 r/T 2 therefore, F c = mv 2 /r or, F c = m4  2 r/T 2

Motion in a Vertical Circle A B FwFw TATA FwFw TBTB

Vertical circle FwFw TATA FwFw TBTB A B Top of Circle at v min T A = 0 and F w = F c therefore, T A + mg = mv 2 /r because T A = 0, mg = mv 2 /r and v 2 = rg

Vertical Circle FwFw TATA FwFw TBTB A B Bottom of Circle v max at bottom thereore, T B - mg = mv 2 /r or F c = T B - F w or T B = mv 2 /r + mg

Cornering on the Horizontal When an object is caused to travel in a circular path because of the force of friction, then,... F c = F F car FwFw FNFN F

Cornering on the Horizontal F c = F F car FwFw FNFN F Therefore, mv 2 /r =  F N because F N = F w = mg, mv 2 /r =  mg … or,  = v 2 /rg

Cornering on a Banked Curve car FwFw FNFN FpFp

Cornering on a Banked Curve car FwFw FNFN FcFc FwFw FNFN FcFc Note how F N is the Resultant 

FwFw FNFN FcFc  If we want to know the angle the curve has to be at to allow the car to circle without friction, then we have to analyze the forces acting on the car. Sin  = F c /F N and, F c = Sin  F N