PRELIMINARY STAGE 2024
Contest 37
Winneba SHS: 53 points (Winner)
Armed Forces SHTS: 43 points
Vakpo SHS: 22 points
Contest 38
St. Joseph’s Seminary SHS: 26 points (Winner)
Suhum Senior High Technical School: 22 points
Breman Asikuma SHS: 18 points
Contest 39
Ghana National College: 66 points (Winner)
Nkwatia Presby SHS: 46 points
Nafana Presby SHS: 15 points
FIRST SET OF QUESTIONS
PREAMBLE
Find the angular frequency of an ideal resonant LC circuit with the given values of L and C.
1. L equals 16 millihenry and C equals 9.0 millifarad
ANSWER: 83 rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{L \times C}}$
$L \times C = 16\times10^{-3} \times 9.0\times10^{-3}$
$L \times C = 16 \times 9\times10^{-3} \times10^{-3}$
$L \times C = 144\times10^{-6}$
$\sqrt{L \times C} = \sqrt{144\times10^{-6}}$
$\sqrt{L \times C} = 12\times10^{-3}$
$\omega = \dfrac{1}{12\times10^{-3}}$
$\omega = \dfrac{1000}{12}$
$\omega \approx 83\text{ rad/s}$
2. L equals 25 millihenry and C equals 64 millifarad.
ANSWER: 25 rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{L \times C}}$
$L \times C = 25\times10^{-3} \times 64\times10^{-3}$
$L \times C = 1600\times10^{-6}$
$\sqrt{L \times C} = \sqrt{1600\times10^{-6}}$
$\sqrt{L \times C} = 40\times10^{-3}$
$\omega = \dfrac{1}{40\times10^{-3}}$
$\omega = \dfrac{1000}{40}$
$\omega = 25\text{ rad/s}$
3. L equals 4 millihenry and C equals 25 millifarad.
ANSWER: 100 rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{L \times C}}$
$L \times C = 4\times10^{-3} \times 25\times10^{-3}$
$L \times C = 100\times10^{-6}$
$\sqrt{L \times C} = \sqrt{100\times10^{-6}}$
$\sqrt{L \times C} = 10\times10^{-3}$
$\omega = \dfrac{1}{10\times10^{-3}}$
$\omega = \dfrac{1000}{10}$
$\omega = 100\text{ rad/s}$
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PRACTICE QUESTIONS
1. QUESTION: Find the natural angular frequency of an ideal LC circuit with $L = 4.0$ mH and $C = 4.0$ mF.
ANSWER: 250 rad/s or $2.5\times10^{2}$ rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{LC}}$
$LC = 4.0\times10^{-3} \times 4.0\times10^{-3}$
$LC = 4\times10^{-3} \times 4\times10^{-3}$
$LC = 4 \times 4 \times10^{-6}$
$LC = 16\times10^{-6}\text{ s}^2$
$\sqrt{LC} = 4\times10^{-3}\text{ s}$
$\omega = \dfrac{1}{4\times10^{-3}}$
$\omega \approx 2.5\times10^{2}\text{ rad/s}$
2. QUESTION: Find the natural angular frequency of an ideal LC circuit with $L = 3.0$ mH and $C = 3.0$ mF.
ANSWER: 330 rad/s or $3.3\times10^{2}$ rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{LC}}$
$LC = 3.0\times10^{-3} \times 3.0\times10^{-3}$
$LC = 3\times10^{-3} \times 3\times10^{-3}$
$LC = 3 \times 3 \times10^{-6}$
$LC = 9\times10^{-6}\text{ s}^2$
$\sqrt{LC} = 3\times10^{-3}\text{ s}$
$\omega = \dfrac{1}{3\times10^{-3}}$
$\omega \approx 3.3\times10^{2}\text{ rad/s}$
3. QUESTION: Find the natural angular frequency of an ideal LC circuit with $L = 6.0$ mH and $C = 6.0$ mF.
ANSWER: 170 rad/s or $1.7\times10^{2}$ rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{LC}}$
$LC = 6.0\times10^{-3} \times 6.0\times10^{-3}$
$LC = 6\times10^{-3} \times 6\times10^{-3}$
$LC = 6 \times 6 \times10^{-6}$
$LC = 36\times10^{-6}\text{ s}^2$
$\sqrt{LC} = 6\times10^{-3}\text{ s}$
$\omega = \dfrac{1}{6\times10^{-3}}$
$\omega \approx 1.7\times10^{2}\text{ rad/s}$
4. QUESTION: Find the natural angular frequency of an ideal LC circuit with $L = 7.0$ mH and $C = 7.0$ mF.
ANSWER: 140 rad/s or $1.4\times10^{2}$ rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{LC}}$
$LC = 7.0\times10^{-3} \times 7.0\times10^{-3}$
$LC = 7\times10^{-3} \times 7\times10^{-3}$
$LC = 7 \times 7 \times10^{-6}$
$LC = 49\times10^{-6}\text{ s}^2$
$\sqrt{LC} = 7\times10^{-3}\text{ s}$
$\omega = \dfrac{1}{7\times10^{-3}}$
$\omega \approx 1.4\times10^{2}\text{ rad/s}$
5. QUESTION: Find the natural angular frequency of an ideal LC circuit with $L = 9.0$ mH and $C = 9.0$ mF.
ANSWER: 110 rad/s or $1.1\times10^{2}$ rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{LC}}$
$LC = 9.0\times10^{-3} \times 9.0\times10^{-3}$
$LC = 9\times10^{-3} \times 9\times10^{-3}$
$LC = 9 \times 9 \times10^{-6}$
$LC = 81\times10^{-6}\text{ s}^2$
$\sqrt{LC} = 9\times10^{-3}\text{ s}$
$\omega = \dfrac{1}{9\times10^{-3}}$
$\omega \approx 1.1\times10^{2}\text{ rad/s}$
6. QUESTION: Find the natural angular frequency of an ideal LC circuit with $L = 12$ mH and $C = 12$ mF.
ANSWER: 83 rad/s or $8.3\times10^{1}$ rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{LC}}$
$LC = 12\times10^{-3} \times 12\times10^{-3}$
$LC = 144\times10^{-6}\text{ s}^2$
$\sqrt{LC} = 12\times10^{-3}\text{ s}$
$\omega = \dfrac{1}{12\times10^{-3}}$
$\omega \approx 83\text{ rad/s}$
7. QUESTION: Find the natural angular frequency of an ideal LC circuit with $L = 1.5$ mH and $C = 1.5$ mF.
ANSWER: 670 rad/s or $6.7\times10^{2}$ rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{LC}}$
$LC = 1.5\times10^{-3} \times 1.5\times10^{-3}$
$LC = 15\times10^{-4} \times 15\times10^{-4}$
$LC = 15 \times 15 \times10^{-8}$
$LC = 2.25\times10^{-6}\text{ s}^2$
$\sqrt{LC} = 1.5\times10^{-3}\text{ s}$
$\omega = \dfrac{1}{1.5\times10^{-3}}$
$\omega \approx 6.7\times10^{2}\text{ rad/s}$
8. QUESTION: Find the natural angular frequency of an ideal LC circuit with $L = 1.2$ mH and $C = 1.2$ mF.
ANSWER: 830 rad/s or $8.3\times10^{2}$ rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{LC}}$
$LC = 1.2\times10^{-3} \times 1.2\times10^{-3}$
$LC = 12\times10^{-4} \times 12\times10^{-4}$
$LC = 12 \times 12 \times10^{-8}$
$LC = 1.44\times10^{-6}\text{ s}^2$
$\sqrt{LC} = 1.2\times10^{-3}\text{ s}$
$\omega = \dfrac{1}{1.2\times10^{-3}}$
$\omega \approx 8.3\times10^{2}\text{ rad/s}$
9. QUESTION: Find the natural angular frequency of an ideal LC circuit with $L = 2.0$ mH and $C = 8.0$ mF.
ANSWER: 250 rad/s or $2.5\times10^{2}$ rad/s
SOLUTION:
$\omega = \dfrac{1}{\sqrt{LC}}$
$LC = 2.0\times10^{-3} \times 8.0\times10^{-3}$
$LC = 2\times10^{-3} \times 8\times10^{-3}$
$LC = 2 \times 8 \times10^{-6}$
$LC = 16\times10^{-6}\text{ s}^2$
$\sqrt{LC} = 4\times10^{-3}\text{ s}$
$\omega = \dfrac{1}{4\times10^{-3}}$
$\omega \approx 2.5\times10^{2}\text{ rad/s}$