Problem 16 IYPT 2027: 16. Magnetic carousel

Problem Description & Analysis

Several neodymium magnets are attached to a horizontal disc, with adjacent magnets oriented in opposite directions. A fixed non-magnetic plate is placed horizontally above the disc. When the disc rotates about its vertical axis and a steel ball is placed on the plate, the ball may move either in the same direction as, or opposite to, the rotation of the disc. Investigate this phenomenon and determine how the motion depends on relevant parameters.

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شار مغناطیسی B (T) 0.3 T

رسانایی σ (MS/m) 35 MS/m

شعاع دیسک R (mm) 80 mm

ضخامت دیسک h (mm) 5 mm

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Python Code Lab

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Python — Editable

# آزمایش «چرخ‌فلک مغناطیسی» — گشتاور ترمزی برحسب سرعت زاویه‌ای
import numpy as np
import matplotlib.pyplot as plt

# --- بخش ۱: داده‌های شما ---
# سرعت زاویه‌ای دیسک آلومینیمی (rad/s)
omega = np.array([5, 10, 20, 40, 60, 80, 100, 140])
# گشتاور ترمزی اندازه‌گیری‌شده (mNm) از نیروسنج
tau_mNm = np.array([2.5, 5.1, 10.3, 20.7, 30.9, 41.5, 51.8, 72.6])

# --- بخش ۲: نظریه: τ = K ω  (خطی)  →  فیت خطی ---
coef = np.polyfit(omega, tau_mNm, 1)
K_fit, offset = coef
tau_th = K_fit*omega + offset

# --- بخش ۳: نمودار ---
fig, ax = plt.subplots(figsize=(8, 4.6))
ax.plot(omega, tau_mNm, 'ro', ms=9, label='measured')
w = np.linspace(0, 150, 100)
ax.plot(w, K_fit*w + offset, 'b-', lw=2,
        label=f'τ = ({K_fit:.4f})·ω + {offset:.2f}')
ax.set_xlabel('ω (rad/s)'); ax.set_ylabel('τ (mNm)')
ax.set_title('Eddy-current brake — torque vs ω')
ax.legend(); ax.grid(True, alpha=0.3)
plt.tight_layout()

# --- بخش ۴: نتایج ---
print(f"Fitted drag coefficient K = {K_fit*1e-3:.5f} N·m·s/rad")
# check τ/ω is constant
ratio = tau_mNm/omega
print(f"τ/ω values (should be constant): mean={ratio.mean():.4f}, std={ratio.std():.4f}")
# Predicted vs measured
print("n ω(rad/s)  τ_meas  τ_fit  residual")
for w, tm, tt in zip(omega, tau_mNm, tau_th):
    print(f"   {w:5.1f}   {tm:5.2f}   {tt:5.2f}   {tm-tt:+.2f}")
print(f"nR² = {1 - np.var(tau_mNm-tau_th)/np.var(tau_mNm):.4f}")

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Problem Description & Analysis

Interactive Simulation (Python)

Python Code Lab

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