A fundamental law of motion states that the acceleration of an object is directly proportional to the resultant force exerted on the object and inversely proportional to the mass.if the proportionality constant has no dimensions, determine the dimensions of forces?

To determine the dimensions of force, let's analyze the given information.

According to the fundamental law of motion, the acceleration (a) of an object is directly proportional to the resultant force (F) exerted on the object and inversely proportional to the mass (m). Mathematically, this can be written as:

a ∝ F/m

Since the proportionality constant has no dimensions, we can write the equation as:

a = k * (F/m)

Where k is the dimensionless constant of proportionality.

Now, let's analyze the dimensions of each term in this equation.

- The acceleration (a) is measured in units of length per time squared. For example, meters per second squared (m/s²).

- The force (F) is what we're trying to determine. Let's denote the dimensions of force as [F].

- The mass (m) is measured in units of mass, such as kilograms (kg).

Now, when we divide force (F) by mass (m), the dimensions of the right-hand side of the equation become:

[F]/[m] = (units of force)/(units of mass) = [F]/[kg]

Since we know that the acceleration (a) has dimensions of length per time squared (m/s²), we can equate the dimensions on both sides of the equation:

(a) = k * [F]/[kg]

Here, we see that the dimensions of force ([F]) must be:

[F] = [kg] * (m/s²)

Therefore, the dimensions of force are mass times length per time squared.