Answered

Which one of the following equations is dimensionally

homogeneous? Show your proof.

AF = ma
BF = m(V2/R)
CF(t2 - t1) = m(V2 – V1)
DF =mV
EF = m(V2 – V1)/(t2 – t1)

where

F = force (N)

m = mass (kg)

a = acceleration (m/s2)

V = velocity (m/s)

R = radius (m)

t = time (s)

Verified Answer :

[F] = [M]*[L]\[T]\[T]

F unit of force
M unit of mass
L unit of length
T unit of time

Explanation:

We need to calculate the equations is dimensionally  homogeneous.

We know that,

Unit of force is Newton.

\(1 N = kg-m/s^2\)

Unit of mass is kg.

Unit of acceleration is m/s².

Unit of velocity is m/s.

Unit of radius is m.

Unit of time is sec.

(a). Given that,

\(F = ma\)

Put the unit of all elements

\(N=kg-m/s^2\)

\(Kg-m/s^2=kg-m/s^2\)

Here, \(N = kg-m/s^2\)

This equations is dimensionally  homogeneous.

(b). Given that,

\(F=\dfrac{mv^2}{r}\)

Put the  unit in the formula

\(N=\dfrac{kg-m^2/s^2}{m}\)

\(kg-m/s^2=kg-m/s^2\)

This equations is dimensionally  homogeneous.

(c). Given that,

\(F(t_{2}-t_{1})=m(v_{2}-v_{1})\)

Put the  unit in the formula

\(N-sec=kg-m/s\)

\(kg-m/s^2\times sec=kg-m/s\)

\(kg-m/s=kg-m/s\)

This equations is dimensionally  homogeneous.

(d). Given that,

\(F = mv\)

Put the  unit in the formula

\(N=kg-m/s\)

\(kg-m/s^2=kg-m/s\)

This equations is not dimensionally  homogeneous.

(e). Given that,

\(F=\dfrac{m(v_{2}-v_{1})}{t_{2}-t_{1}}\)

Put the  unit in the formula

\(N=\dfrac{kg-m/s}{s}\)

\(kg-m/s^2=kg-m/s^2\)

This equations is dimensionally  homogeneous.

Hence, This is required solution.

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