Question

The contripetal force F acting on a particle moving uniformly in a circle may depend upon mass (m), velocity (v) and redio ( r) of the circle . Derive the formula for F using the method of dimensions.

Answer 1

Answer:

f = mv^2/r

Explanation:

f= mass * accelerartion

thus, f= MLT^-2

so let us say,

f= MLT^-2 = (M^x)(V^y)(L^z)  ....................(i)

from (i), we can say that x=1

we know that V^2=(L^2)(T^-2)

thus, f= MLT^-2 = M(V^2)(L^-1) = (M^X)(V^Y)(L^Z)

therefore, x=1, y=2 and z=-1

thus

f=mv^2/r

P.S. the brackets r used here for understanding and proper reading purpose only, they are not used in original terminology

Answer 2

GIVEN :

The centripetal force F depends upon mass (m), velocity (v) and radius ( r) of the circle .

TO FIND :

Derive the formula for F using the method of dimensions.

SOLUTION :

◆We know,

Force ,F= ma

m - mass , a- acceleration

◆And thus,F= MLT^-2 ,

in dimensional analysis , m - M ,

a - LT^-2

◆Here, velocity,V = LT^-1

◆So, writing force in terms of velocity

And checking their dimensions,

◆MLT^-2 = (M^x)(V^y)^2(L^z)  

◆ By further solving ,

= (M^1)(LT^-1)^2(L^-1)

◆So equating both sides , we get

x = 1, y = 2 ,z = -1

◆Thus equation becomes,

Centripetal force F=(mv^2)/r

ANSWER :

Centripetal force F=(mv^2)/r