The density of air in atomsphere decreases with height and can be expressed by the relation.
`rho=rho_(0)e^(-Ah)`
Where `rho_(0)` is the density at sea-level, A is a constant and h is the height. Calculate the atmospheric pressure at sea -level. assume g to be constant. `9.8m//s^(2), rho_(0) =1.3 kg//m^(3)` and `A=1.2xx10^(-4) m^(-1)`

(a). It is known that density rho of air decreases with height y as rho=rho_(0)e^(-y//y(_o)) ltb rgt where p_(o)=1.25kgm^(-3) is the density at sea level, and y_(o) is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of g remains constant (b). A large He balloon of volume 1425m^(3) is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise? [take y_(o)=8000m and rho_(He)=0.18kgm^(-3) ]

The density inside a solid sphere of radius a is given by rho=rho_(0)(1-r/a) , where rho_(0) is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre.

Estimate the pressure deep inside the sea at a depth h below the surface. Assume that the density fo water is rho_(0) at sea level and its bulk modulus is B. P_(0) is the atmosphere pressure at sea level P is the pressure at depth 'h'

A spherical region of space has a distribution of charge such that the volume charge density varies with the radial distance from the centre r as rho=rho_(0)r^(3),0<=r<=R where rho_(0) is a positive constant. The total charge Q on sphere is

(a) Pressure decreases as one ascends the atmosphere. If the density of air is rho, What is the change in pressure dp over a differential height dh ? (b) Considering the pressure p to be proportional to the density, find the pressure p at a height h if the pressure on the sureface of the earth is p_0. (c ) If p_0 = 1.03 xx 10^5 Nm^(-2), rho_0 = 1.29kg m^(-3) and g = 9.8 ms^(-2), at what height will the pressure drop to (1//10) the value at the surface of the earth ? (d) This model of the atmosphere works for relatively small distance. Identify the underlying assumption that limits the model.

The linear mass density i.e. mass per unit length of a rod of length L is given by rho = rho_(0)(1 + (x)/(L)) , where rho_(0) is constant , x distance from the left end. Find the total mass of rod and locate c.m. from the left end.

If the density of a gas at the sea level at 0^(@)C is 1.29 kg m^(-3) , what is its molar mass? (Assume that pressure is equal to 1"bar" .)

A thin rod AB of length a has variable mass per unit length rho_(0) (1 + (x)/(a)) where x is the distance measured from A and rho_(0) is a constant. Find the mass M of the rod.