A uniform board of length L is sliding along a smooth (frictionless) horizontal

Question

A uniform board of length L is sliding along a smooth (frictionless) horizontal plane as in the figure below. The board then slides across the boundary with a rough horizontal surface. The coefficient of kinetic friction between the board and the second surface is mu k. Find the acceleration of the board at the moment its front end has traveled a distance x beyond the boundary. (Use any variable or symbol stated above along with the following as necessary: g.) a = in the direction opposite the motion The board stops at the moment its back end reaches the boundary, as in the figure above. Find the initial speed (v) of the board. (Use any variable or symbol stated above along with the following as necessary: g.) v =

Explanation / Answer

a) so x part is on rough surface. means mx/L mass is on rough surface.

so f = (mx/L)g

ma = (mx/L)g

a = xg/L ....Ans

b) usinh work energy theorem .

K.E. = work done by friction

work done by friction dW = F.dx = (mg/L)xdx

integrating it from x = 0 to x = L

W = (mg/L)x2 /2 |L0  

W = (mg/L)L2 /2 = mgL/2

so, mv2 /2 =   mgL/2   

v = gL .........Ans

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