A cylinder rolls without slipping on a horizontal plane. The radius of curvature is r. Find the curvature radii of trajectories traced out by the points A and B in the figure. 
 
 
 
 
 

 

Asked by Sourav | 8th May, 2015, 12:28: PM

Expert Answer:

 
 
At point A:
 
Velocity at A = Velocity due to linear motion of the cylinder + Rotating velocity of the cylinder (both with magnitude v)
Velocity at A = v + v = 2v
The radial acceleration(centripetal acceleration) experienced at point A will be begin mathsize 12px style straight v squared over straight r end style
begin mathsize 12px style The space radii space of space curvature space traced space by space the space point space straight A space can space be space obtained space by colon straight R subscript 1 space equals space fraction numerator open parentheses 2 straight v close parentheses squared over denominator normal space acceleration end fraction equals fraction numerator 4 straight v squared over denominator begin display style straight v squared over straight r end style end fraction straight R subscript 1 space equals 4 space straight r end style
 
 
At point B: 
 
Velocity at B, 
 
begin mathsize 12px style straight v subscript straight B equals square root of straight v squared plus straight v squared end root space space space space equals v square root of 2 end style
The tangential velocity vector makes an angle 45º with the vertical. The ardial vector of the accelertion also makes an angle 45º with the vertical.
begin mathsize 12px style Therefore space the space radial space acceleration space equals space straight v squared over straight r cos space 45 degree space equals fraction numerator straight v squared over denominator straight r square root of 2 end fraction Therefore space the space radial space accelerationexperienced space at space point space straight B space will space be space fraction numerator straight v squared over denominator straight r square root of 2 end fraction The space radii space of space curvature space of space the space trakjectory space traced space by space the space point space straight B space will space be colon straight R subscript 2 space equals space fraction numerator open parentheses straight v square root of 2 close parentheses squared over denominator fraction numerator straight v squared over denominator straight r square root of 2 end fraction end fraction straight R subscript 2 space equals 2 square root of 2 space straight r end style

Answered by Jyothi Nair | 11th May, 2015, 11:01: AM