S.I. units：練習問題

The intensity \(I\) of a wave can be expressed as \(I = 2\pi^2 f^2 A^2 \rho v\), where \(f\) is the frequency, \(A\) is the amplitude, \(\rho\) is the density of the medium, and \(v\) is the speed of the wave. Which of the following correctly shows the S.I. base units for each side of the equation, confirming its homogeneity?

A technician measures the potential difference across a resistor as \(V = 12.0 \pm 0.2\text{ V}\) and the current as \(I = 2.00 \pm 0.05\text{ A}\). The resistance is calculated using \(R = V/I\). In terms of S.I. base units, what is the value and absolute uncertainty of the resistance \(R\)?

Which of the following units is equivalent to the S.I. base units for the quantity pressure?

The Planck constant \(h\) has units of \(\text{J s}\). Which of the following expressions represents the S.I. base units for the Planck constant?

The resistive force \(F\) acting on a sphere of radius \(r\) moving at speed \(v\) through a fluid with dynamic viscosity \(\eta\) is given by Stokes' Law: \(F = 6\pi \eta r v\). What are the S.I. base units for dynamic viscosity \(\eta\)?

Show that the S.I. base units for the Young modulus \(E\) are \(\text{kg m}^{-1}\text{ s}^{-2}\). Recall that \(E = \frac{\sigma}{\epsilon}\), where stress \(\sigma = \frac{F}{A}\) and strain \(\epsilon = \frac{\Delta L}{L}\).

A force \(F\) is given by the equation \(F = kv^2\), where \(v\) is velocity. Determine the S.I. base units of the constant \(k\) and state whether the equation is homogeneous if \(F\) is measured in Newtons and \(v\) in \(\text{m s}^{-1}\).

(a) The drag force \( F \) acting on an object falling through a fluid is given by the equation \( F = \frac{1}{2} C_D \rho A v^2 \), where \( C_D \) is the drag coefficient (dimensionless), \( \rho \) is the density of the fluid, \( A \) is the cross-sectional area, and \( v \) is the velocity. Use S.I. base units to check the homogeneity of this equation.
(b) A student measures the diameter of a spherical ball bearing as \( (12.0 \pm 0.1) \text{ mm} \) and its mass as \( (7.50 \pm 0.05) \text{ g} \). Calculate the density of the material in \( \text{kg m}^{-3} \) and determine the absolute uncertainty in this value.