The wave velocity (v) inside water depends on the wavelength `(lambda)` of the wave the density (`sigma`) of water and acceleration due to gravity (g). Work out the relation among the quantities from dimensional analysis.

For a planet the escape velocity (v) of a body depends upon the radius ( R) of the planet and acceleration due to gravity (g) . Establish a relation among the above physical quantities.

The velocity (v ) of sound in a gas depends on the pressure (p) and density ( sigma ) of the gas. Using dimensional analysis establish a relation between v,p and sigma .

The errors in the measurement of the effective length ( l ) and the time period (T) of a pendulum executing simple harmonic motion are 1% and 2% respectively. Estimate the error in the determination of acceleration due to gravity (g) using this pendulum. Given the relation among the relevant quantities, T = 2pi sqrt((l)/(g)).

A hemi-spherical tank of radius 2 m is initially full of water and has an outlet of 12c m^2 cross-sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law v(t)=0.6sqrt(2gh(t)), where v(t) and h(t) are, respectively, the velocity of the flow through the outlet and the height of water level above the outlet and the height of water level above the outlet at time t , and g is the acceleration due to gravity. Find the time it takes to empty the tank.

A drop of oil rises through water with an acceleration alphag . If alpha is a constant quantity and g is the acceleration due to gravity, find the specific gravity of the oil. Neglect the friction of water.