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The distance of a geo-stationary satellite from the centre of the earth (Radius R = 6400 km) is nearest to -





(b)    6R from the surface of earth and 7R from the centre.

In order to make the effective acceleration due to gravity equal to zero at the equator, the angular velocity of rotation of the earth about its axis should be (g=10 ms-2 and radius of earth is 6400 kms)





(b)  g'=g-ω2Rcos2λ 

For weightlessness at equator λ=0 and g'=0

0=g-ω2Rω=gR=1800rads

 A simple pendulum has a time period T1 when on the earth’s surface and T2 when taken to a height R above the earth’s surface, where R is the radius of the earth. The value of is T2/T1 is 





(d) If acceleration due to gravity is at the surface of earth then at height R its value becomes g'=gRR+h2=g4

T1=2πlg  and T2=2πlg/4  

T2T1=2

A body of mass m is taken from earth surface to the height h equal to radius of earth, the increase in potential energy will be





(b) U=mgh1+hR=12mgR (h=R)

Periodic time of a satellite revolving above Earth’s surface at a height equal to R, radius of Earth, is
[g is acceleration due to gravity at Earth’s surface]





(b)     T=2πR+h3gR2=2π2R3gR2=42πRg

An artificial satellite moving in a circular orbit around the earth has a total (kinetic + potential) energy E0 . Its potential energy is         





(c) Potential energy = 2 x (Total energy) = 2E0

Because we know = U=-GMmr and E0=-GMm2r 

Given radius of Earth ‘R’ and length of a day ‘T’ , the height of a geostationary satellite is [G–Gravitational Constant, M–Mass of Earth] 





(c)     T=2πr3GMT2=4π2GMR+h3R+h=GMT24π213h=GMT24π213-R

A rocket of mass M is launched vertically from the surface of the earth with an initial speed V. Assuming the radius of the earth to be R and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is   





(c) K.E.=U

12MV2=GMeM1R-1R+h                 ...(i)

Also      g=GMeR2                                     ...(ii)

On solving (i) and (ii)   h=R2gRV2-1

Two bodies of masses m1 and m2  are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance r between them is





(b) Let velocities of these masses at r distance from each other be v1 and v2 respectively. By conservation of momentum 

m1v1-m2v2=0

m1v1=m2v2                                                          ...(i)

By conservation of energy 

change in P.E.= change in K.E.

Gm1m2r=12m1v12+12m2v22

m12v12m1+m22v22m2=2Gm1m2r                                     ...(ii)

On solving equation (i) and (ii) 

v1=2Gm22r(m1+m2)      and    v2=2Gm12r(m1+m2)           

 

 νapp=v1+v2=2Gr(m1+m2)

The rotation period of an earth satellite close to the surface of the earth is 83 minutes. The time period of another earth satellite in an orbit at a distance of three earth radii from its surface will be





(c) For first satellite and r1=R and T1=83minutes

For second satellite r2=4R

  T2=T1r2r13/2=T143/2=8T1=8×83=664 minutes 

A projectile is projected with velocity kve in vertically upward direction from the ground into the space. ( ve is escape velocity and k<1). If air resistance is considered to be negligible then the maximum height from the centre of earth to which it can go, will be : (R = radius of earth)





(c)Loss in Kinetic energy = Gain in Potential energy

12m(kve)2=mgh1+hR12mk22gR=mgh1+hR h=Rk21-k2

Height of projectile from the earth's surface = h

Height from the centre r=R+h=R+Rk21-k2

By solving r=R1-k2

A satellite of mass m is circulating around the earth with constant angular velocity. If radius of the orbit is R0 and mass of the earth M, the angular momentum about the centre of the earth is





(a) Angular momentum = Mass ×Orbital velocity ×Radius

=m×GMR0×R0=mGMR0

If the distance between the earth and the sun becomes half its present value, the number of days in a year would have been





(b) According to Kepler's third law, the ratio of the squares of the periods of any two planets revolving about the sun is equal to the ratio of the cubes of their average distances from the sun i.e.

T1T22=r1r23=r112r13=8 T1T2=22 T2=T122=365 days22=129 days

According to Kepler, the period of revolution of a planet (T) and its mean distance from the sun (r) are related by the equation





(b) T2r3=constantT2r3=constant

A geostationary satellite orbits around the earth in a circular orbit of radius 36000 km. Then, the time period of a satellite orbiting a few hundred kilometres above the earth’s surface (Rearth= 6400 km) will approximately be -





(c) T2T1=r2r13/2T2=246400360003/22 hour