Dimensions of Force

To determine the dimensions of force using area, velocity, and density as the fundamental quantities, we can start by analyzing the equation for force. In the International System of Units (SI), force is defined as the product of mass and acceleration:

F = ma

In this equation, F represents force, m represents mass, and a represents acceleration. To express force in terms of area, velocity, and density, we need to relate mass and acceleration to these fundamental quantities.

Relating Mass to Density

Density is defined as the mass per unit volume, given by the equation:

ρ = m / V

Here, ρ represents density, m represents mass, and V represents volume. Rearranging this equation, we can express mass in terms of density and volume:

m = ρV

Relating Acceleration to Area and Velocity

Acceleration can be expressed in terms of velocity and time using the equation:

a = Δv / Δt

Here, Δv represents the change in velocity, and Δt represents the change in time. To relate acceleration to area and velocity, we need to find a relationship between velocity, time, and area.

Using the definition of velocity as the change in displacement over time:

v = Δs / Δt

Here, Δs represents the change in displacement. If we assume the area is a square with sides of length L, then the displacement can be related to the area as:

Δs = L

Substituting this relationship into the equation for velocity:

v = L / Δt

Now, we can substitute this expression for velocity into the equation for acceleration:

a = Δv / Δt = (v - u) / Δt = (L / Δt - 0) / Δt = L / (Δt)²

Final Expression for Force

Finally, we can substitute the expressions for mass and acceleration into the equation for force:

F = ma = (ρV)(L / (Δt)²) = ρVL / (Δt)²

Dimensions of Force

From the equation above, we can determine the dimensions of force in terms of area, velocity, and density:

Force = [ρVL / (Δt)²]

- The dimensions of density are mass per unit volume, [M L⁻³].
- The dimensions of area are [L²].
- The dimensions of velocity are [L T⁻¹].
- The dimensions of time are [T].

Combining these dimensions, we find the dimensions of force as:

[M L T⁻²]