A balloon with mass m is descending down with an acceleration a (where a < g). How much mass should be removed from it so that it starts moving up with an acceleration a ?

A system consists of three masses m₁, m₂ and m₃ connected by a string passing over a pulley P. The mass m₁ hangs freely and m₂ and m₃ are on a rough horizontal table (the coefficient of friction = $$\mu $$). The pulley is frictionless and of negligible mass. The downward acceleration of mass m₁ is (Assume m₁ = m₂ = m₃ = m)

A car is moving in a circular horizontal track of radius 10 m with a constant speed of 10 m/s. A bob is suspended from the roof of the car by a light wire of length 1.0 m. The angle made by the wire with the vertical is

The upper half of an inclined plane of inclination $$\theta $$ is perfectly smooth while lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom, if the coefficient of friction between the block and lower half of the plane is given by