Length of a year on a planet is the duration in which it completes one revolution around the sun. Assume path of the planet known as orbit to be circular with sun at the center. The length T of a year of a planet orbiting around the sun in circular orbit depends on universal gravitational constant G, mass m of the sun and radius r of the orbit.
$If T \propto G^{a} m^{b} r^{c}, the value of a + b + 2 c is __________$

Question

Length of a year on a planet is the duration in which it completes one revolution around the sun. Assume path of the planet known as orbit to be circular with sun at the center. The length T of a year of a planet orbiting around the sun in circular orbit depends on universal gravitational constant G, mass m of the sun and radius r of the orbit.

$If T \propto G^{a} m^{b} r^{c}, the value of a + b + 2 c is __________$

Infinity Learn · Accepted Answer

[T]$$=$$[G]amb[r]$$cM0L0T=M$$−$$1L3T$$−$$2a$$[M]b[L]$$c=Mb$$−$$aL3a+cT$$−$$2a$$⇒ $$a=$$−$$12$$,$$b=$$−$$12$$,$$c=32$$⇒ $$a+b+2c=2$$

Dimensions of physical quantities

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$\begin{array}{l} [T] = [G]^{a} {[m]}^{b} [r]^{c} \\ M^{0} L^{0} T = {[M^{- 1} L^{3} T^{- 2}]}^{a} [M]^{b} [L]^{c} = M^{b - a} L^{3 a + c} T^{- 2 a} \\ \Rightarrow a = - \frac{1}{2}, b = - \frac{1}{2}, c = \frac{3}{2} \\ \Rightarrow a + b + 2 c = 2 \end{array}$

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[T]=[G]amb[r]cM0L0T=M−1L3T−2a[M]b[L]c=Mb−aL3a+cT−2a⇒ a=−12,b=−12,c=32⇒ a+b+2c=2

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$\begin{array}{l} [T] = [G]^{a} {[m]}^{b} [r]^{c} \\ M^{0} L^{0} T = {[M^{- 1} L^{3} T^{- 2}]}^{a} [M]^{b} [L]^{c} = M^{b - a} L^{3 a + c} T^{- 2 a} \\ \Rightarrow a = - \frac{1}{2}, b = - \frac{1}{2}, c = \frac{3}{2} \\ \Rightarrow a + b + 2 c = 2 \end{array}$