The collar moves without friction along the horizontal rod w

The collar moves without friction along the horizontal rod, which rotates about the vertical rod at constant angular velocity as shown. Initially when r = 0.5 m, the slider has an outer speed of 3 m/s with respect to the horizontal rod. Since no radial forces are exerted on the collar, its radial acceleration a_r = 0 Determine the collar\'s velocity in polar coordinates when r = 0.6 m.

Solution

assume

that at A :rA=0.5 m and at B : rB=0.6

given Va=3 m/s

according to Law of conservation of momentum

For the constant angular velocity

angular momentum at A= angular momentum at B

m*rA*Va=m*rB*Vb

Va*rA=Vb*rB

3(0.5)=Vb(0.6)

Vb=3(0.5)/(0.6)

Vb=2.5 m/s

here the velocity is along the horizontal direction whiche is along x- axis assume

so the polar coordinates are Vb=2.5 i m/s

Also the rod rotates with constant angular velocity w= 6 rad/s

so the velocity at B is Vby

Vby=rB* w

Vby=0.6*6

Vby=3.6 j m/s (in the counter clockwise direction so

Vby=-3.6 j

therefore the polar coordinates of the collar at B

V=Vbx+Vby

V=2.5 i-3.6 j m/s