Question

what is the answer of this physics Question​

Answer 1

Answer is given in the pic.

Answer 2

Question:-

Given that :-

$\rm \int \dfrac{dy}{ \sqrt{2ay - y} } = a {}^{m } \sin {}^{ - 1} ( \dfrac{y - b}{b} )$

where b is constant. using dimensional analysis. find the value of m

Solution:-

we have

$\rm : \implies \int \dfrac{dy}{ \sqrt{2ay - y} } = a {}^{m } \sin {}^{ - 1} ( \dfrac{y - b}{b} )$

Concept is use in dimensional analysis, we use

$\rm \: : \implies \: numerical \: value \: \times unit \: = constant$

Now

$: \implies \rm \: n_1u_1 = n_2u_2$

$: \implies \rm \: n_2 = n_1 \bigg[ \dfrac{M_1}{M_2} \bigg]{}^{a} \bigg[ \dfrac{L_1}{ L_2} \bigg] {}^{b} \bigg[ \dfrac{T_1}{T_2} \bigg] {}^{c}$

Where a , b and c are dimenstions of mass , length and time in given quantity

we get

$\rm : \implies \int \dfrac{dy}{ \sqrt{2ay - y} }$

So dimensions of integration is

$\rm : \implies \: [M ] {}^{0} [L] {}^{0} [T] {}^{0}$

dimensions of constant = 1

we can write as

$\rm : \implies \: [M ] {}^{0} [L] {}^{0} [T] {}^{0} = {a}^{m} \times 1$

So value of m is

$: \implies\rm \: m = 0$