本题目来源于试卷: 2007美国US F=MA物理竞赛，类别为 美国F=MA物理竞赛
[单选题]
Find the period of smallu :kewf0hk1x) oscillations of a water pogo, which is a ddz;e)w e4/ng stick of mass gdden e)/w ;z4$m$ in the shape of a box (a rectangular parallelepiped.) The stick has a length $L$, a width $w$ and a height $h$ and is bobbing up and down in water of density $\rho$. Assume that the water pogo is oriented such that the length $L$ and width $w$ are horizontal at all times. Hint: The buoyant force on an object is given by $F_{buoy}=\rho Vg$, where $V$ is the volume of the medium displaced by the object and $\rho$ is the density of the medium. Assume that at equilibrium, the pogo is floating.
A. $2\pi\sqrt{L/g}$
B. $\pi\sqrt{\rho w^2 L^2 g / (mh^2)}$
C. $2\pi\sqrt{mh^2 / (\rho L^2 w^2 g)}$
D. $2\pi\sqrt{m/(\rho wLg)}$
E. $\pi\sqrt{m/(\rho wLg)}$
参考答案: D
本题详细解析:
This is a simple harmonic motion problem.
1. At equilibrium: The buoyanti66iy t x lj+.pry+pv( force $F_B$ equals the weight $mg$. Let $y_0$ be the submerged depth.
The submerged volume is $V_{eq} = L w y_0$.
$F_B = \rho V_{eq} g = \rho (Lw y_0) g$.
So, $mg = \rho Lw y_0 g$.
2. Displaced system: Push the pogo down by a small distance $y$.
The new submerged depth is $(y_0 + y)$.
The new buoyant force is $F_B' = \rho V_{new} g = \rho (Lw (y_0 + y)) g = \rho Lw y_0 g + \rho Lw y g$.
3. Restoring force: The net force $F_{net}$ on the pogo is $F_B' - mg$ (upwards).
$F_{net} = (\rho Lw y_0 g + \rho Lw y g) - mg$.
Since $mg = \rho Lw y_0 g$ (from equilibrium), these terms cancel:
$F_{net} = (\rho Lw g) y$.
This is the restoring force, which acts in the opposite direction to the displacement $y$.
So, $F_{restore} = - (\rho Lw g) y$.
4. SHM equation: This is in the form $F = -k_{eff} y$, where the effective spring constant is $k_{eff} = \rho Lw g$.
The period $T$ of a mass-spring system is $T = 2\pi\sqrt{m/k_{eff}}$.
Substituting our $k_{eff}$: $T = 2\pi\sqrt{m / (\rho Lw g)}$.
(Note: The height $h$ is irrelevant as long as the pogo floats).
| 
