Answer to Question #166205 in Physics for gibson mkandawire

Question

1.Use the principle of homogeneity, check the correctness of the following equation

ℎ=2𝑇𝐶𝑜𝑠𝛽/𝑟𝑑𝑔

where β is the angle of contact, d is the density of a liquid, r is the radius of the tube, g is acceleration due to gravity, h is the height of the liquid and T is the surface tension.

2. If the escape velocity depends upon;

(i) Acceleration due to gravity of the planet

(ii) The radius of the planet

Establish the relationship between them by dimensional analysis.

[Hint, 𝑣𝑒𝛼 𝑔𝑎𝑅𝑏]

Answer 1

1)

$$[L]=\frac{[MT^{-2}]}{[ML^{-3}][L][LT^{-2}]}\\ [L]=[L]$$

So, the relation is correct.

2)

$$v=g^aR^b\\ [v]=[g]^a[R]^b\\LT^{-1}=(LT^{-2})^a(L)^b\\a=0.5,b=0.5\\ v=k\sqrt{Rg}$$