The force \(F\) is represented by equation \(F=P \ell+Q \ell^{-1}\), where \(\ell\) is the length. The unit of \(P\) is same as that of $\ldots \ldots \ldots \ldots \ldots$ (a) Surface tension (b) velocity (c) force (d) momentum

The force equation can be written as: \(F = P \cdot ℓ + Q \cdot ℓ^{-1}\) Now let's break down each part of the equation.

Force is a vector quantity that measures the interaction between two objects. The SI unit for force is the newton (N).

Length is a scalar quantity that measures the distance between two points. The SI unit for length is the meter (m).

P and Q are constants. The unit of P needs to be determined, while the unit of Q is not required for this exercise. To determine the unit of P, we will focus on the term Pℓ in the force equation.

By examining the term Pℓ in the force equation, we can determine the unit of P: \(F = P \cdot ℓ + Q \cdot ℓ^{-1}\) Since force (F) is in newtons (N) and length (ℓ) is in meters (m), we need to find a unit for P such that the product of P and ℓ has the same unit as force (N). Now let's analyze the given options: (a) Surface Tension: The unit of surface tension is N/m. Since we have Pℓ, multiplying surface tension by length (ℓ) gives us: \((N/m) \cdot m = N\) (b) Velocity: The unit of velocity is m/s. If we multiply velocity by length (ℓ), it gives us: \((m/s) \cdot m = m^2/s\) (c) Force: The unit of force is N. Multiplying force by the length (ℓ) yields: \(N \cdot m\) (d) Momentum: The unit of momentum is kg∙m/s. If we multiply momentum by length (ℓ), we get: \((kg \cdot m/s) \cdot m = kg \cdot m^2/s\) Comparing units obtained from the provided options, the unit in option (a) is the only one that matches the requirements we identified for Pℓ to have the same unit as force (N). Therefore, the unit of P is the same as that of surface tension. Correct answer: (a) Surface tension