Question

A small ball rolls around a horizontal circle at height y inside a frictionless hemispherical bowl of radius R, as shown in the figure (Figure 1) with constant speed. Find the speed of the ball in terms of y, R, and g. A block of mass m is at rest at the origin at t = 0. It is pushed with constant force F0 from .x = 0 to x = L across a horizontal surface whose coefficient of kinetic friction is mu k = mu 0 (1 - x/L). Find the expression for the block's speed as it reaches position L

Answer 1

Apply 2nd law; net force (applied - friction) equal ma;
ma = Fo - f = Fo - umg = Fo - uo(1 - x/L)mg
a = (Fo/m) - uo(1-x/L)g

a = dv/dt , but you want to express this in terms of x ,rather then t, so use the chain rule;
dv/dt = (dv/dx)(dx/dt) = v(dv/dx)

so your acceleration eq is;
v(dv/dx) = (Fo/m) - uo(1-x/L)g

to find v , multiply thru the dx and integrate the left side from 0 to v and the right side from 0 to x;
vdv = (Fo/m)dx - uogdx + (uog/L)xdx

Integrate
(1/2)v^2 = (Fo/m)x - uogx + (uog/2L)x^2

You can do the rest. Just solve for v in terms of x , then evaluate it at x=L if you want the velocity at that location.