Due to an explosion underneath water, a bubbl | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given, the time period of oscillation $T$ is $T \propto p^\alpha S^\beta E^\gamma$ or $T = k p^\alpha S^\beta E^\gamma$.Substituting the dimension

Vedclass · Accepted Answer

$\alpha=-\frac{5}{6}, \beta=\frac{1}{2}, \gamma=\frac{1}{3}$

Vedclass · Answer

(B) Given, the time period of oscillation $T$ is $T \propto p^\alpha S^\beta E^\gamma$ or $T = k p^\alpha S^\beta E^\gamma$.Substituting the dimensions of $T, p, S$, and $E$:$[M^0 L^0 T^1] = [ML^{-1} T^{-2}]^\alpha [ML^{-3}]^\beta [ML^2 T^{-2}]^\gamma$$[M^0 L^0 T^1] = [M^{\alpha+\beta+\gamma} L^{-\alpha-3\beta+2\gamma} T^{-2\alpha-2\gamma}]$Equating the powers of $M, L$, and $T$ on both sides, we get:$1) \alpha + \beta + \gamma = 0$$2) -\alpha - 3\beta + 2\gamma = 0$$3) -2\alpha - 2\gamma = 1$From equation $(3)$, $\alpha + \gamma = -\frac{1}{2}$.Substituting this into equation $(1)$, $-\frac{1}{2} + \beta = 0 \implies \beta = \frac{1}{2}$.Now, substitute $\beta = \frac{1}{2}$ into equation $(2)$:$-\alpha - 3(\frac{1}{2}) + 2\gamma = 0 \implies -\alpha + 2\gamma = \frac{3}{2}$.We have the system:$i) \alpha + \gamma = -\frac{1}{2}$$ii) -\alpha + 2\gamma = \frac{3}{2}$Adding $(i)$ and $(ii)$, $3\gamma = 1 \implies \gamma = \frac{1}{3}$.Substituting $\gamma = \frac{1}{3}$ into $(i)$, $\alpha + \frac{1}{3} = -\frac{1}{2} \implies \alpha = -\frac{1}{2} - \frac{1}{3} = -\frac{5}{6}$.Thus, $\alpha = -\frac{5}{6}, \beta = \frac{1}{2}, \gamma = \frac{1}{3}$.

Similar Questions

If $\text{force} = \frac{\alpha}{\text{density} + \beta^3}$,then the dimensional formulae of $\alpha$ and $\beta$ are respectively:

What is the dimensional formula of $ab^{-1}$ in the equation $(P+\frac{a}{V^2})(V-b)=RT$,where letters have their usual meaning?

In a nuclear power plant,the energy released depends on the mass of the uranium sample $(m)$,the length of the oscillator $(\ell)$,and the frequency $(f)$ of oscillation as $E = m^{x} \ell^{y} f^{z}$. Then,what is the value of $x + y + z$?

If $x = a/t + b/t^2 + c$,where $x$ is the distance travelled by the body in metres and $t$ is the time in seconds,then the units of $b$ are:

If time $(t)$,velocity $(u)$,and angular momentum $(I)$ are taken as the fundamental units,then the dimension of mass $(m)$ in terms of $(t)$,$(u)$,and $(I)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module