If a pushing force making an angle @ with horizontal is applied on a block of mass m placed on a horizontal table and angle of friction is b then the minimum magnitude of force required to move the block is what???

							
Look at the figures is attachments. Here FF is the force applied and ff is force due to friction
F(cos α)=f cos β F (cos α)= f coβ (1)
Also,
f = μ N f = μ N (2)
where, μμ is the coefficient of friction and NN is the normal reaction by the floor on the block.
By force balance on the block along the vertical direction we get
N= F sin α + f sin β + mg N=F sin α +f sin β + mg (3)

subtracting 2 and 3 we get
N = F sin α + mg ( 1 − μ sin β ) N = F sin α + mg ( 1 − μ sin β) (4)

Now subtracting 2 and 4 we get
F cos α = μ F sin α + mg ( 1 − μ sin β )  cos β F cos α = μ F sin α + mg ( 1 − μ sin β ) cos β

F cos α ( 1− μ sin β ) =μ cos β ( F sin α + mg ) F cos α ( 1 – μ sin β ) = μ cos β ( F sin α + mg )

F cos α ( 1 − μ sin β ) = F μ cos β sin α + μ mg cos β F cos α ( 1−μ sin β) =F μ cos β  sin α+ μ mg cos β

F [cos α− μ cos α sin β− μ cos β sin α ]= μ mg cos β F [ cos α –μ cos α sin β− μ cos β sin α ] = μ mg cos β

F = μ mg cos β cos α – μ ( cos α sin β + cos β sin α ) F = μ mg cos β cos α – μ ( cos α sin β + cos β sin α )

F = μ mg cos β cos α – μ sin ( α + β ) F = μ mg cos β cos α – μ sin ( α + β )

since sin ( α + β ) = cos α sin β + cos β sin α sin ( α + β ) = cos α sin β + cos β sin α.

F = F min = μ mg cos β cos α – μ sin ( α + β )
F = F min = μ mg cos β cos α – μ sin ( α + β ).