Question

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(sinè-ìcosè)]/(ìsinè+cosè)

Please help! Thanks!

Answer 1

Take x-axis directed along the incline downwards, and y-axis 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).