What should be the mass of a block to prevent it from sliding down a ramp with a coefficient of friction .65, inclined at 30 degree?

For a block placed on an inclined ramp which is not
frictionless, there are two forces acting. One of them is the gravitational force of
attraction that makes the block slide down the ramp and a counteracting force due to
friction.

The ramp in the problem is inclined at an angle
of 30 degrees. The force with which it is pulled down due to gravity is equal to the
product of its mass m and the acceleration due to gravity
g.

We divide this force F = mg, into two components, one
along the ramp in the downward direction which is equal to m*g* cos 60 and one
perpendicular to the ramp equal to m*g* sin 60.

The force
of friction opposing the motion of the block is Mu * N, where Mu is the coefficient of
friction and N is the normal force.

Here the opposing force
is .65*m*g* sin 60. This has to be less than the component of the force moving the block
downward.

This gives 0.65*m*g* sin 60 =< m*g* cos
60

We see that m and g cancel from both the sides and we
are left with

=> 0.65 sin 60 =< cos
60

This implies that whether the block slides down or not
is not decided by its mass, but by the inclination of the slope and the coefficient of
friction. Here 0.65* sin 60 = .5629 is greater than cos 60 =
0.5

Therefore the block does not slide down the ramp
irrespective of its mass.