When a body slides down from rest along a smooth inclined plane making an angle of 30$$^\circ$$ with the horizontal, it takes time T. When the same body slides down from the rest along a rough inclined plane making the same angle and through the same distance, it takes time $$\alpha$$T, where $$\alpha$$ is a constant greater than 1. The co-efficient of friction between the body and the rough plane is $${1 \over {\sqrt x }}\left( {{{{\alpha ^2} - 1} \over {{\alpha ^2}}}} \right)$$ where x = __________.

A car is moving on a plane inclined at 30$$^\circ$$ to the horizontal with an acceleration of 10 ms^$$-$$2 parallel to the plane upward. A bob is suspended by a string from the roof of the car. The angle in degrees which the string makes with the vertical is ______________. (Take g = 10 ms^$$-$$2)

The coefficient of static friction between two blocks is 0.5 and the table is smooth. The maximum horizontal force that can be applied to move the blocks together is .................. N. (take g = 10 ms^-2)

A body of mass 'm' is launched up on a rough inclined plane making an angle of 30$$^\circ$$ with the horizontal. The coefficient of friction between the body and plane is $${{\sqrt x } \over 5}$$ if the time of ascent is half of the time of descent. The value of x is __________.