A sphere of radius [tex]$r$[/tex] is moving at speed [tex]$v$[/tex] through air of density [tex]$\rho$[/tex]. The resistive force [tex]$F$[/tex] acting on the sphere is given by the expression
[tex]$F=B r^2 \rho v^k$[/tex]
where [tex]$B$[/tex] and [tex]$k$[/tex] are constants without units.
i. State the SI base units of [tex]$F, \rho$[/tex] and [tex]$v$[/tex].
ii. Use base units to determine the value of [tex]$k$[/tex].

The SI base units of force F are k g c d o t m c d o t s − 2 .
The SI base units of density ρ are k g c d o t m − 3 .
The SI base units of velocity v are m c d o t s − 1 .
By equating the powers of the base units in the equation F = B r 2 ρ v k , we find that k = 2 .

k = 2 ​
Explanation

Understanding the Problem We are given the equation F = B r 2 ρ v k , where F is the resistive force, r is the radius, ρ is the density, v is the speed, and B and k are dimensionless constants. We need to find the SI base units of F , ρ , and v , and then determine the value of k using base units.

Finding SI Units of Force The SI base units of force F are kilograms times meters per second squared, which is k g ⋅ m ⋅ s − 2 . This comes from Newton's second law, F = ma , where m is mass (kg) and a is acceleration (m/s 2 ).

Finding SI Units of Density The SI base units of density ρ are kilograms per cubic meter, which is k g ⋅ m − 3 . Density is defined as mass per unit volume, and volume is measured in cubic meters (m 3 ).

Finding SI Units of Velocity The SI base units of velocity v are meters per second, which is m ⋅ s − 1 . Velocity is defined as the rate of change of displacement with respect to time.

Expressing the Equation in Base Units Now, we will determine the value of k . We can write the given equation in terms of base units: k g ⋅ m ⋅ s − 2 = ( m ) 2 ( k g ⋅ m − 3 ) ( m ⋅ s − 1 ) k

Simplifying the Equation Simplifying the equation, we get: k g ⋅ m ⋅ s − 2 = k g ⋅ m 2 ⋅ m − 3 ⋅ m k ⋅ s − k = k g ⋅ m 2 − 3 + k ⋅ s − k = k g ⋅ m k − 1 ⋅ s − k

Equating the Powers Equating the powers of the base units, we have: For kg: 1 = 1 For m: 1 = k − 1 For s: − 2 = − k

Solving for k Solving the equation 1 = k − 1 for k , we get k = 1 + 1 = 2 .
Solving the equation − 2 = − k for k , we get k = 2 .
Therefore, the value of k is 2.

Final Answer The SI base units of F are k g ⋅ m ⋅ s − 2 , the SI base units of ρ are k g ⋅ m − 3 , the SI base units of v are m ⋅ s − 1 , and the value of k is 2.


Examples
Understanding the drag force on a sphere moving through a fluid is crucial in many real-world applications. For example, engineers use this principle to design vehicles that move efficiently through air or water, such as airplanes and submarines. By knowing the relationship between the drag force, the sphere's radius, the fluid's density, and the velocity, they can optimize the shape and size of the vehicle to minimize resistance and improve performance. This also applies to designing pipelines and understanding the movement of particles in fluids.

Answered by GinnyAnswer | 2025-07-08