A 0.6 kg bead slides…

A 0.6 kg bead slides on a curved wire, starting from rest at point a which is 5.2 meters on the y axis the x axis is the vertical component.  The segment from point A to point B is frictionless and point B rests on the x-axis.  The segment from point B to Point C is rough and point c is at 0.9meters on the y Axis.  Point C is relative to point B.  Acceleration to gravity is $9.8 m/s^2$. Find the speed of the bead at point B.  Answer in units of m/s

Part 2: If the bead comes to rest at point C, find the change in mechanical energy due to friction as it moves from point B to point C.  Answer in Joules

Ssupposing the Y component of the position is the Height of the bead (the problem does not specify this exactly) then the speed in point B can be found by variation of total energy considerations (there is no friction on the segment AB)

$E(A) = E(B)$

$m*g*y(A) =m*v^2(B)/2$

$v(B) =\sqrt{2*g*y(A)} =\sqrt {2*9.8*5.2} =10.096 m/s$

. The change in total mechanical energy due to friction from B to C is equal to the variation in total energy.

$\Delta E = m*v^2(B)/2 -m*g*y(C) =0.6*10,096^2/2 -0.6*9.8*0.9 =25.287 J$