Density, upthrust and Stokes' law：練習問題

Which of the following expressions is correct for tensile stress?

A small spherical bead falls at a constant terminal velocity \( v \) through a large container of oil with a viscosity \( \eta \). The experiment is repeated with a different oil that has a viscosity of \( 2\eta \).
Assuming the upthrust is negligible and the flow remains laminar, what is the new terminal velocity of the bead?

A small steel sphere of radius \( 1.5 \times 10^{-3} \text{ m} \) falls through a liquid of viscosity \( 0.65 \text{ Pa s} \) with a terminal velocity of \( 0.040 \text{ m s}^{-1} \). Assuming Stokes' law applies, what is the magnitude of the viscous drag force acting on the sphere?

On a force-extension graph for a metal spring, what does the region from the origin up to the limit of proportionality represent?

Two wires, \( X \) and \( Y \), are made of the same metal. Wire \( X \) has length \( L \) and radius \( r \). Wire \( Y \) has length \( 2L \) and radius \( 2r \). When a tensile force \( F \) is applied to wire \( X \), it extends by \( \Delta L \).
What force is required to produce the same extension \( \Delta L \) in wire \( Y \)?

A small steel ball bearing of mass \(m\) and radius \(r\) is dropped from rest into a tall cylinder filled with oil of density \(\rho_{f}\). The density of the steel is \(\rho_{s}\) and the viscosity of the oil is \(\eta\).

(a) Draw a free-body force diagram for the ball bearing as it falls at terminal velocity. Label all forces clearly.
(b) By considering the forces acting on the ball, derive an expression for the terminal velocity \(v_{t}\) in terms of \(r, g, \eta, \rho_{s},\) and \(\rho_{f}\).
(c) In an experiment, a ball of radius 2.0 mm reaches a terminal velocity of \(0.15\text{ m s}^{-1}\). Calculate the viscosity of the oil. (\(\rho_{s} = 7800\text{ kg m}^{-3}\), \(\rho_{f} = 950\text{ kg m}^{-3}\)).