if force F velocity V and time T are taken as fundamental units then the dimensions of mass are?

To determine the dimensions of mass when we consider force (F), velocity (V), and time (T) as fundamental units, we can use Newton's second law of motion, which states that force is equal to mass times acceleration (F = m * a). Acceleration can be expressed in terms of velocity and time. Let's break this down step by step.

Understanding the Relation Between Force, Mass, and Acceleration

According to the equation of motion, acceleration (a) is defined as the change in velocity (V) over time (T):

• a = V / T

Now, substituting this definition of acceleration into Newton's second law gives us:

• F = m * (V / T)

Rearranging the Formula

If we rearrange this equation to solve for mass (m), we can express it in terms of the other quantities:

• m = F * (T / V)

Expressing Dimensions in Fundamental Units

Now we need to express the dimensions of force in terms of our fundamental units. Force is defined as mass times acceleration, so its dimensions can be derived as follows:

• Acceleration has dimensions of length (L) divided by time squared (T²), which can be written as: [a] = [L][T]^-2
• Therefore, the dimensions of force are: [F] = [m][L][T]^-2

Substituting this expression for force back into our rearranged equation for mass:

• m = ([m][L][T]^-2) * (T / V)

Finding the Dimensions of Mass

Now, we need to express velocity (V) in terms of fundamental dimensions. Velocity is length divided by time:

• [V] = [L][T]^-1

We can substitute this into our equation for mass:

• m = ([m][L][T]^-2) * (T / ([L][T]^-1))

Now let's simplify this expression:

• m = ([m][L][T]^-2) * (T^2 / [L])
• m = [m][T]^0

Final Expression for Mass Dimensions

After simplifying, we find that the dimensions of mass (m) can be expressed as:

• [m] = [F][T^2][V^-1]

Therefore, the dimensions of mass in terms of force, velocity, and time are derived as follows:

• [m] = [F][T^2][V]^-1

This shows how mass is fundamentally linked to the other quantities we're considering. By understanding these relationships, you gain a deeper insight into the principles that govern motion and force in physics.