Physics
posted by Anonymous on .
A block of mass m lies on a rough plane which is inclined at an angle è to the horizontal. The coefficient of static friction between the block and the plane is ì. A force of magnitude P is now applied to the block in a horizontal direction, towards the plane.
Show that the minimum value of P which is necessary to ensure that the block remains at rest on the plane, is
P=[mg(sinèìcosè)]/(ìsinè+cosè)
Please help! Thanks!

Take xaxis directed along the incline downwards, and yaxis directed normaly away from the incline.
Projections of the equation of motion on the x and y axes are:
m•g•sinα  P•cosα – F(fr) = 0,
m•g•cos α + N P•sinα = 0.
F(fr) = μ •N = μ •(m•g•cos α+P•sinα).
0 = m•g•sinα  P•cosα  μ •(m•g•cos α+P•sinα) =
= m•g•sinα  P•cosα  μ •m•g•cos α+μ•P•sinα.
P = m•g•( sinα  μ•cos α)/(cos α + μ•sinα).