Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit.

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Q.
Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit. [2024]

1 $T^{2} = \frac{4 π^{2} r^{3}}{G M}$

2 $T^{2} = \frac{4 π^{2} r^{2}}{G M}$

3 $T^{2} = 4 π^{2} r^{3}$

4 $T^{2} = \frac{4 π^{2} r}{G M^{2}}$

Ans.
(1)

According to principle of homogeneity dimension of LHS should be equal to dimensions of RHS. $[T^{2}] = T^{2},$

Dimension of G is $[M^{- 1} L^{3} T^{- 2}]$

Now,

(1) $[\frac{4 π^{2} r^{3}}{G M}] = \frac{L^{3}}{M^{- 1} L^{3} T^{- 2} M^{1}} = T^{2}$

(2) $[\frac{4 π^{2} r^{2}}{G M}] = \frac{L^{2}}{M^{- 1} L^{3} T^{- 2} M^{1}} = L^{- 1} T^{2}$

(3) $[4 π^{2} r^{3}] = L^{3}$

(4) $[\frac{4 π^{2} r}{G M^{2}}] = \frac{L}{M^{- 1} L^{3} T^{- 2} M^{2}} = M^{- 1} L^{- 2} T^{2}$

Solution

Q.
Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit. [2024]

1 $T^{2} = \frac{4 π^{2} r^{3}}{G M}$

2 $T^{2} = \frac{4 π^{2} r^{2}}{G M}$

3 $T^{2} = 4 π^{2} r^{3}$

4 $T^{2} = \frac{4 π^{2} r}{G M^{2}}$

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Solution

Q. Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit. [2024]

1 T2=4π2r3GM

2 T2=4π2r2GM

3 T2=4π2r3

4 T2=4π2rGM2

Ans. (1) According to principle of homogeneity dimension of LHS should be equal to dimensions of RHS. [T2]=T2, Dimension of G is [M-1L3T-2] Now, (1) [4π2r3GM]=L3M-1L3T-2M1=T2 (2) [4π2r2GM]=L2M-1L3T-2M1=L-1T2 (3) [4π2r3]=L3 (4) [4π2rGM2]=LM-1L3T-2M2=M-1L-2T2

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Q.
Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit. [2024]

1 $T^{2} = \frac{4 π^{2} r^{3}}{G M}$

2 $T^{2} = \frac{4 π^{2} r^{2}}{G M}$

3 $T^{2} = 4 π^{2} r^{3}$

4 $T^{2} = \frac{4 π^{2} r}{G M^{2}}$