Question

One of the fundamental laws of motion states that the acceleration of an object is directly proportional to the resultant force on it and inversely proportional to its mass. If the proportionality constant is defined to have no dimensions, determine the dimensions of force.
(b) The newton is the SI unit of force. According to the results for (a), how can you express a force having units of newtons by using the fundamental units of mass, length, and time?

Answer 1

Mass has units of kg.....kilogram (mass)
Acceleration has units of m/s^2......m is meters (length) and s is seconds (time)
1 Newton (unit of force) is 1 kg . m/s^2

Answer 2

What you just stated:\[a \propto F \text{ and } a \propto \dfrac{1}{m}\]Therefore:\[a\propto \dfrac{F}{m}\]\[a = C\dfrac{F}{m} \text{ where } C\in\mathbb R\]\[F = \dfrac{ma}{C} \text{ where } C\in\mathbb R\]Since C is dimensionless, it does not contribute to the dimensions of the force.\[[\text{Newtons}]=\dfrac{\left[\text{kilograms}\right]\dfrac{\left[\text{meters}\right]}{\left[\text{seconds}\right]^2}}{\left[1\right]}\]\[\left[\text{Newtons}\right]=\dfrac{\left[\text{kilograms}\right]\left[\text{meters}\right]}{\left[\text{seconds}\right]^2}\]Thus \[1\text{N}=1\dfrac{\text{kg}\cdot\text{m}}{\text{s}^2}\]