JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Gravitation

Two satellites A and B of masses 200 kg and 400 kg are revolving round the earth at height of 600 km and 1600 km respectively.
If

T_{A}

and

T_{B}

are the time periods of

A

and

B

respectively then the value of

T_{B} - T_{A}

:
[ Given : radius of earth

= 6400 km

, mass of earth

= 6 \times 1 0^{24} kg

]

Answer: (B)

Solution

Orbital period depends on the orbital radius alone, so the two masses play no part. Measuring from the centre of the earth,

r_{A} = 6400 + 600 = 7000

km and

r_{B} = 6400 + 1600 = 8000

km.

Balancing gravity against the centripetal requirement,

\frac{GM m}{r ^{2}} = \frac{4 π ^{2} m r}{T ^{2}}

, gives

T = 2 π \frac{r ^{3}}{GM}, GM = (6.67 \times 1 0^{- 11}) (6 \times 1 0^{24}) = 4.0 \times 1 0^{14} m^{3} s^{- 2}

T_{A} = 2 π \frac{( 7 \times 1 0 ^{6} ) ^{3}}{4.0 \times 1 0 ^{14}} = 5.8 \times 1 0^{3} s, T_{B} = 2 π \frac{( 8 \times 1 0 ^{6} ) ^{3}}{4.0 \times 1 0 ^{14}} = 7.1 \times 1 0^{3} s

The satellite further out lags behind by

T_{B} - T_{A} \approx 1.3 \times 1 0^{3}

JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Single correct option