Question

The distance moved by a particle in time from centre of ring under the influence of its

gravity is given by x = a sint where a and are constants. Ifis found to depend on

the radius of the ring (r), its mass (m) and universal gravitation constant (G),find using

dimensional analysis an expression for  in terms of r, m and G​

Answer 1

$\bold{\mathff{\omega =\sqrt{\dfrac{GM}{r^3}}}}$

Given,

The quantity W depends on

(i) Universal Gravitational Constant G

(ii) radius (r)

(iii) mass (m)

[G] = [M^(-1) L^3T^(-2)]

[r] = L

[m] = M

$\bold{\large\omega}$ = r^(x) m^(y) G^(z)

$\bold{\large\omega}$ = L^(x) M^(y) [M^(-1) L^3 T^(-2)]^z

T^(-1) = L^(x) M^(y) M^(-z) L^(3z) T^(-2z)

T^(-1) = M^(y-z) L^(x+3z) T^(-2z)

equating both sides, We get ,

x = 3/2

z = 1/2

y = 1/2

Substituting values in main equation, we get,

$\bold{\mathff{\omega =\sqrt{\dfrac{GM}{r^3}}}}$

Answer 2

Answer with explanation: