The force acting on a body is given as where  is displacement and t is time. If the dimensions of  is  then the value of  is ________ .

To solve the problem, let's follow the structured approach.

1. Break Down the Problem

We need to find the value of a parameter based on the relationship between force, displacement, and time given in the problem. The dimensions of displacement are provided, and we are required to deduce other parameters.

2. Relevant Concepts

The force $F$ is related to displacement $s$ and time $t$. The dimension of force is given by: $[F] = [M][L][T^{-2}]$ where $[M]$ is mass, $[L]$ is length, and $[T]$ is time.

We are told that the dimensions of displacement $s$ are: $[s] = [L]$

3. Analysis and Detail

If we denote the unknown value as $k$ such that $F$ can be expressed in terms of displacement and time with the dimensions of $s$ and $t$, we have: $F = k \cdot s^a \cdot t^b$ Now we substitute the dimensions: $[F] = [M][L][T^{-2}] \quad \text{and} \quad [s] = [L], \quad [t] = [T]$ Thus, $[M][L][T^{-2}] = k \cdot [L]^a \cdot [T]^b$ To maintain dimensional homogeneity, we can equate the exponents of $[L]$ and $[T]$ on both sides.

Analyzing the dimensions, we obtain:

• For $[L]$: $1 = a \quad \text{(from the length dimension)}$
• For $[T]$: $-2 = b \quad \text{(from the time dimension)}$

4. Verify and Summarize

From our analysis, we have determined that $a = 1$ and $b = -2$. This leads us to conclude that $k$ is dimensionless, thus it can take the value of 1 or any scalar since it does not affect the dimensional consistency of the equation as written.

Final Answer

The value of $k$ in this relationship is $1$.