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].
Answer (2)
The SI base units of force F are kg m s − 2 .
The SI base units of density ρ are kg m − 3 .
The SI base units of velocity v are m s − 1 .
By substituting the SI base units into the equation F = B r 2 ρ v k and equating the powers, we find that k = 2 .
k = 2
Explanation
Problem Analysis We want to find the SI base units of F , ρ , and v , and then determine the value of k in the equation F = B r 2 ρ v k , where B is a dimensionless constant.
SI Units of Force The SI base units of force F can be derived from Newton's second law, F = ma , where m is mass (kg) and a is acceleration (m s − 2 ). Therefore, the SI base units of force are kg m s − 2 .
SI Units of Density The SI base units of density ρ can be derived from the definition of density, ρ = V m , where m is mass (kg) and V is volume (m 3 ). Therefore, the SI base units of density are kg m − 3 .
SI Units of Velocity The SI base units of velocity v can be derived from the definition of velocity, v = t d , where d is distance (m) and t is time (s). Therefore, the SI base units of velocity are m s − 1 .
Substituting SI Units into the Equation Now, we substitute the SI base units into the given equation F = B r 2 ρ v k :
kg m s − 2 = (m) 2 (kg m − 3 ) (m s − 1 ) k Simplifying, we get: kg m s − 2 = kg m 2 − 3 + k s − k = kg m k − 1 s − k
Solving for k Equating the powers of the base units, we have: For kg: 1 = 1 For m: 1 = k - 1 For s: -2 = - k From the equation for s, we have k = 2 . From the equation for m, we have 1 = k - 1, so k = 2 .
Final Answer Therefore, the value of k is 2.
Examples
Understanding the dimensions of physical quantities and how they relate to each other is crucial in many engineering applications. For example, when designing a bridge, engineers need to understand how the force applied to the bridge relates to the materials used and the dimensions of the bridge. Similarly, in fluid dynamics, understanding the relationship between force, density, velocity, and dimensions is essential for designing efficient aircraft or ships. Dimensional analysis ensures that equations are consistent and can be used to predict the behavior of physical systems.
The SI base units for force ( F ) are kg m s − 2 , for density ( ρ ) are kg m − 3 , and for velocity ( v ) are m s − 1 . When analyzing the equation for resistive force, it is determined that the value of k is 2. Thus, we conclude that k = 2 .
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