Problem 7 IYPT 2027: 7. Sinking funnel

Problem Description & Analysis

When a long-stemmed and sufficiently heavy funnel is released vertically into water with the stem pointing downward, it undergoes a series of damped vertical oscillations while gradually sinking. Investigate the motion and stability of the sinking funnel and the conditions under which the funnel tips over.

Related Images

Interactive Simulation (Python)

This simulation runs directly in your browser — no Python installation needed. Initial loading may take a few seconds.

طول لولهٔ قیف L (m) 0.15 m

نصف زاویهٔ قیف (°) 30 °

شعاع بالای قیف R (m) 0.06 m

جرم قیف m (kg) 0.05 kg

Press the button below to start.

Python Code Lab

Edit this starter code, enter your own experimental data, and run it directly in the browser — no Python installation required.

Python — Editable

# آزمایش قیف در حال غرق — دورهٔ نوسان برحسب زاویه
import numpy as np
import matplotlib.pyplot as plt

g = 9.81
rho_w = 1000.0

# --- بخش ۱: داده‌های شما ---
# با چند قیف با نصف‌زاویهٔ مختلف
theta_deg = np.array([15, 25, 35, 45, 55, 65])
# دورهٔ متوسط نوسان (s) از میانگین ۱۰ نوسان
T_meas = np.array([0.62, 0.68, 0.74, 0.82, 0.92, 1.05])
# پارامترهای ثابت
R = 0.06
L = 0.15
m = 0.05

# --- بخش ۲: نظریه ---
def period(theta_rad):
    h = np.linspace(0.001, L*0.95, 2000)
    r_wl = np.clip(R - h*np.tan(theta_rad), 1e-4, None)
    Vd = (np.pi/3)*h*(R**2 + R*r_wl + r_wl**2)
    idx = np.argmin(np.abs(Vd - m/rho_w))
    h0 = h[idx]; r0 = R - h0*np.tan(theta_rad)
    Aw = np.pi*r0**2
    k  = rho_w*g*Aw
    m_add = rho_w * (np.pi/3)*h0*(R**2+R*r0+r0**2) * 0.5
    return 2*np.pi*np.sqrt((m+m_add)/k)

T_th = np.array([period(np.radians(t)) for t in theta_deg])

# --- بخش ۳: نمودار ---
fig, ax = plt.subplots(figsize=(8, 4.6))
th_smooth = np.linspace(10, 75, 60)
T_smooth  = [period(np.radians(t)) for t in th_smooth]
ax.plot(th_smooth, T_smooth, 'b-', lw=2, label='theory')
ax.plot(theta_deg, T_meas, 'ro', ms=9, label='measured')
ax.set_xlabel('half-angle θ (deg)'); ax.set_ylabel('period T (s)')
ax.set_title('Sinking-funnel bobbing period')
ax.legend(); ax.grid(True, alpha=0.3)
plt.tight_layout()

# --- بخش ۴: نتایج ---
print(" θ(°)   T_th(s)  T_meas(s)  Δ(%)")
for t, tt, tm in zip(theta_deg, T_th, T_meas):
    print(f"  {t:2.0f}    {tt:5.3f}    {tm:5.3f}    {(tm-tt)/tt*100:+5.1f}")
rms = np.sqrt(np.mean(((T_meas-T_th)/T_th)**2))*100
print(f"nRMS relative error = {rms:.2f}%")

Ready to run

Output