Question

Which one of the following equations is dimensionally
homogeneous? Show your proof.
a. F = ma
b. F = m(V2/R)
c. F(t2 - t1) = m(V2 – V1)
d. F =mV
e. F = 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)

Answer 1

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

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

Answer 2

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.