Find Dimension of Mass in V T F System JEE Main

❓ Question

Velocity $(V)$, Time $(T)$, and Force $(F)$ are taken as fundamental quantities.

Find the dimensions of mass $(m)$ in terms of $V$, $T$, and $F$.

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✍️ Solution

Assume,

$$m\propto V^aT^bF^c$$

Therefore,

$$[M]=[V]^a[T]^b[F]^c$$

We know,

$$[V]=[LT^{-1}]$$
$$[T]=[T]$$

and

$$[F]=[MLT^{-2}]$$

Substituting,

$$[M]=[LT^{-1}]^a[T]^b[MLT^{-2}]^c$$

On simplifying,

$$[M]=[M^cL^{a+c}T^{-a+b-2c}]$$

Comparing powers of $M,L,T$,

$$c=1$$
$$a+c=0$$
$$-a+b-2c=0$$

Since $c=1$,

$$a+1=0$$
$$a=-1$$

Now,

$$-(-1)+b-2=0$$
$$1+b-2=0$$
$$b=1$$

Thus,

$$[m]=[V^{-1}TF]$$

✅ Final Answer

$$\boxed{[m]=[FTV^{-1}]}$$