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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???
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???

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3 years ago

```							Look at the figures is attachments. Here FF is the force applied and ff is force due to frictionF(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 getN= F sin α + f sin β + mg N=F sin α +f sin β + mg (3)subtracting 2 and 3 we getN = F sin α + mg ( 1 − μ sin β ) N = F sin α + mg ( 1 − μ sin β) (4)Now subtracting 2 and 4 we getF 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 βF 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 βF [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 βF = 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 ).
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3 years ago
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