OCR MEI
Further Mechanics Minor
2023
June
— Question 1
Exam Board
OCR MEI
Module
Further Mechanics Minor (Further Mechanics Minor)
Year
2023
Session
June
Topic
Dimensional Analysis
1
State the dimensions of the following quantities.
Force
Velocity
Density
A student investigating the drag force \(F\) experienced by an object moving through air conjectures the formula
\(\mathrm { F } = \mathrm { ku } ^ { 2 } \left( \rho \mathrm {~m} ^ { 2 } \right) ^ { \frac { 1 } { 3 } }\),
where
\(k\) is a dimensionless constant
\(u\) is the air velocity relative to the moving object
\(\rho\) is the air density
\(m\) is the mass of the object.
Show that the student's formula is dimensionally consistent.
The student carries out experiments in an airflow tunnel. When the air density is doubled, the drag force is found to double as well, with all other conditions remaining the same.
Show that the student's formula is inconsistent with the experimental observation.
The student's teacher suggests revising the formula as
\(\mathrm { F } = \mathrm { k } \rho ^ { \alpha } \mathrm { u } ^ { \beta } \mathrm { A } ^ { \gamma }\)
where \(m\) has been replaced by \(A\), the cross-sectional area of the object. The constant \(k\) is still dimensionless.
Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\).