A student uses dimensional analysis to analyze the time ($au$) it takes for a capacitor (C) to discharge through a resistor (R). They correctly identify that $au$ depends on R and C. However, they are unable to derive the precise relationship $au = RC$. What limitation of dimensional analysis explains this?

If the formula for the velocity of a wave is given by $v = \sqrt{\frac{T}{\mu}}$, where $T$ is tension and $\mu$ is mass per unit length. Using dimensional analysis, which aspect of this formula cannot be derived?

Can dimensional analysis derive equations involving trigonometric functions?

Which of these is a limitation of dimensional analysis?

In a vernier callipers, $21$ divisions on the vernier scale coincide with $20$ divisions on the main scale. If each division on the main scale is $0.5$ mm, find the vernier constant (in cm).

Dimensional analysis can help in deriving relationships between physical quantities. However, it cannot:

Dimensional analysis is used to check the validity of an equation relating the speed of sound (v) in a gas to its pressure (P) and density ($\rho$). Which limitation prevents dimensional analysis from distinguishing between $v = \sqrt{P/\rho}$ and $v = \sqrt{14P/\rho}$?

A student uses dimensional analysis to analyze the time ($au$) it takes for a capacitor (C) to discharge through a resistor (R). They correctly identify that $au$ depends on R and C. However, they are unable to derive the precise relationship $au = RC$. What limitation of dimensional analysis explains this?

Dimensional analysis cannot be used to derive relationships containing:

The relative density of the material of a body is the ratio of its weight in air and the loss of its weight in water. By using a spring balance, the weight of the body in air in measured to be 5.00 $\pm$ 0.05N. The weight of the body in water is measured to be 4.00 $\pm$ 0.05N. Then the maximum possible percentage error in relative density is

Dimensional analysis can be used to: