Energy (E) is expressed in terms of mass (m), distance (x) and time (t) as E=a x+(t2 √b/m). If the dimensions of (b/a) is [ M p L q T r], then p+q+r=.

Question

Energy (E) is expressed in terms of mass (m), distance (x) and time (t) as E=a x+(t2 √b/m). If the dimensions of (b/a) is [ M p L q T r], then p+q+r=. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

From principle of homogeneity, [ a ]=([ E ]/[ x ])=([ M 1 L 2 T -2]/ L 1)=[ M 1 L 1 T -2] Also [√ b ] =([ m ][ E ]/[ t ]2) =([ M 1][ M 1 L 2 T -2]/[ T 2]) =[ M 2 L 2 T -4] ∵ [b] =[ M 4 L 4 T -8] ∴ ([b]/[a])=([M4 L4 T-8]/[M1 LL T-2]) =[M3 L3 T-6]=[MP Lq Tr] ∴ p +q+ r=3+3-6=0

Q. Energy (E) is expressed in terms of mass (m), distance (x) and time (t) as E=ax+mt2b​​. If the dimensions of ab​ is [MpLqTr], then p+q+r=________.

From principle of homogeneity, [a]=[x][E]​=L1[M1L2T−2]​=[M1L1T−2] Also [b​]=[t]2[m][E]​ =[T2][M1][M1L2T−2]​ =[M2L2T−4] ∵[b]=[M4L4T−8] ∴[a][b]​=[M1LLT−2][M4L4T−8]​ =[M3L3T−6]=[MPLqTr] ∴p+q+r=3+3−6=0

Q. Energy $(E)$ is expressed in terms of mass $(m)$ , distance $(x)$ and time $(t)$ as $E = a x + \frac{t ^{2} b}{m}$ . If the dimensions of $\frac{b}{a}$ is $[M^{p} L^{q} T^{r}]$ , then $p + q + r =$ ________.