PRELIMINARY STAGE 2026
Contest 22
Wesley Grammar School: 46 points
Kofi Agyei SHTS: 41 points
Yilo Krobo SHS: 27 points
Contest 23
Adiembra SHS: 54 points
T.I. AMASS, Kumasi: 46 points
PRESEC, Bechem: 29 points
ROUND 2 - SPEED RACE
QUESTION
Determine the moment of inertia of a sphere of mass 16.0 kg and radius 0.200 m about a diameter.
ANSWER: $I = 0.256\text{ kg}\cdot\text{m}^2$
SOLUTION 1:
$m = 16\text{ kg}$
$r = 2\times10^{-1}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 16 \times (2\times10^{-1})^2$
$I = \dfrac{32}{5} \times 4\times10^{-2}$
$I = 64\times10^{-1} \times 4\times10^{-2}$
$I = 256\times10^{-3}\text{ kg}\cdot\text{m}^2$
$I = 0.256\text{ kg}\cdot\text{m}^2$ (since the least sf in the question is 3)
SOLUTION 2:
$r^2 = 2\times10^{-1} \times 2\times10^{-1}$
$r^2 = 4\times10^{-2}\text{ m}^2$
$I = \dfrac{2}{5} \times m \times r^2$
$I = 4\times10^{-1} \times 16 \times 4\times10^{-2}$
$I = 256\times10^{-3}\text{ kg}\cdot\text{m}^2$
$I = 0.256\text{ kg}\cdot\text{m}^2$ (since the least sf in the question is 3)
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PRACTICE QUESTIONS
1. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 12.0 kg and radius 15.0 cm about a diameter.
ANSWER: 0.108 kg·m² or $1.08\times10^{-1}$ kg·m²
SOLUTION:
$r = 15.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 12.0 \times (15.0\times10^{-2})^2$
$I = 0.108\text{ kg}\cdot\text{m}^2$
2. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 12.0 kg and radius 10.0 cm about a diameter.
ANSWER: 0.0480 kg·m² or $4.80\times10^{-2}$ kg·m²
SOLUTION:
$r = 10.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 12.0 \times (10.0\times10^{-2})^2$
$r^2 = 0.0100\text{ m}^2$
$I = \dfrac{2 \times 12.0 \times 0.0100}{5}$
$I = \dfrac{2 \times 12 \times 10^{-2}}{5}$
$I = 0.0480\text{ kg}\cdot\text{m}^2$
3. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 25.0 kg and radius 40.0 cm about a diameter.
ANSWER: 1.60 kg·m²
SOLUTION:
$r = 40.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 25.0 \times (40.0\times10^{-2})^2$
$r^2 = 0.160\text{ m}^2$
$I = \dfrac{2 \times 25.0 \times 0.160}{5}$
$I = \dfrac{2 \times 25 \times 16\times10^{-2}}{5}$
$I = 2 \times 5 \times 16 \times10^{-2}$
$I = 16 \times10^{-1}$
$I = 1.60\text{ kg}\cdot\text{m}^2$
4. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 25.0 kg and radius 18.0 cm about a diameter.
ANSWER: 0.324 kg·m² or $3.24\times10^{-1}$ kg·m²
SOLUTION:
$r = 18.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 25.0 \times (18.0\times10^{-2})^2$
$I = 0.324\text{ kg}\cdot\text{m}^2$
5. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 8.00 kg and radius 30.0 cm about a diameter.
ANSWER: 0.288 kg·m² or $2.88\times10^{-1}$ kg·m²
SOLUTION:
$r = 30.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 8.00 \times (30.0\times10^{-2})^2$
$r^2 = 0.0900\text{ m}^2$
$I = \dfrac{2 \times 8.00 \times 0.0900}{5}$
$I = \dfrac{2 \times 8 \times 9\times10^{-2}}{5}$
$I = 0.288\text{ kg}\cdot\text{m}^2$
6. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 4.00 kg and radius 20.0 cm about a diameter.
ANSWER: 0.0640 kg·m² or $6.40\times10^{-2}$ kg·m²
SOLUTION:
$r = 20.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 4.00 \times (20.0\times10^{-2})^2$
$r^2 = 0.0400\text{ m}^2$
$I = \dfrac{2 \times 4.00 \times 0.0400}{5}$
$I = \dfrac{2 \times 4 \times 4\times10^{-2}}{5}$
$I = 0.0640\text{ kg}\cdot\text{m}^2$
7. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 2.50 kg and radius 15.0 cm about a diameter.
ANSWER: 0.0225 kg·m² or $2.25\times10^{-2}$ kg·m²
SOLUTION:
$r = 15.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 2.50 \times (15.0\times10^{-2})^2$
$r^2 = 0.0225\text{ m}^2$
$I = \dfrac{2 \times 2.50 \times 0.0225}{5}$
$I = \dfrac{2 \times 25\times10^{-1} \times 225\times10^{-4}}{5}$
$I = 2 \times 5 \times 225 \times10^{-5}$
$I = 225 \times10^{-4}$
$I = 0.0225\text{ kg}\cdot\text{m}^2$
8. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 7.50 kg and radius 16.0 cm about a diameter.
ANSWER: 0.0768 kg·m² or $7.68\times10^{-2}$ kg·m²
SOLUTION:
$r = 16.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 7.50 \times (16.0\times10^{-2})^2$
$r^2 = 0.0256\text{ m}^2$
$I = \dfrac{2 \times 7.50 \times 0.0256}{5}$
$I = \dfrac{2 \times 75\times10^{-1} \times 256\times10^{-4}}{5}$
$I = 2 \times 15 \times 256 \times10^{-5}$
$I = 3 \times 256 \times10^{-4}$
$I = 0.0768\text{ kg}\cdot\text{m}^2$
9. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 5.00 kg and radius 50.0 cm about a diameter.
ANSWER: 0.500 kg·m² or $5.00\times10^{-1}$ kg·m²
SOLUTION:
$r = 50.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 5.00 \times (50.0\times10^{-2})^2$
$r^2 = 0.250\text{ m}^2$
$I = \dfrac{2 \times 5.00 \times 0.250}{5}$
$I = \dfrac{2 \times 5 \times 25\times10^{-2}}{5}$
$I = 2 \times 25 \times10^{-2}$
$I = 0.500\text{ kg}\cdot\text{m}^2$
10. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 15.0 kg and radius 12.0 cm about a diameter.
ANSWER: 0.0864 kg·m² or $8.64\times10^{-2}$ kg·m²
SOLUTION:
$r = 12.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 15.0 \times (12.0\times10^{-2})^2$
$r^2 = 0.0144\text{ m}^2$
$I = \dfrac{2 \times 15.0 \times 0.0144}{5}$
$I = \dfrac{2 \times 15 \times 144\times10^{-4}}{5}$
$I = 2 \times 3 \times 144 \times10^{-4}$
$I = 0.0864\text{ kg}\cdot\text{m}^2$
11. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 16.0 kg and radius 25.0 cm about a diameter.
ANSWER: 0.400 kg·m² or $4.00\times10^{-1}$ kg·m²
SOLUTION:
$r = 25.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 16.0 \times (25.0\times10^{-2})^2$
$r^2 = 0.0625\text{ m}^2$
$I = \dfrac{2 \times 16.0 \times 0.0625}{5}$
$I = \dfrac{2 \times 16 \times 625\times10^{-4}}{5}$
$I = 2 \times 16 \times 125 \times10^{-4}$
$I = 0.400\text{ kg}\cdot\text{m}^2$
12. QUESTION: Determine the moment of inertia of a uniform solid sphere of mass 2.50 kg and radius 24.0 cm about a diameter.
ANSWER: 0.0576 kg·m² or $5.76\times10^{-2}$ kg·m²
SOLUTION:
$r = 24.0\times10^{-2}\text{ m}$
$I = \dfrac{2}{5}mr^2$
$I = \dfrac{2}{5} \times 2.50 \times (24.0\times10^{-2})^2$
$r^2 = 0.0576\text{ m}^2$
$I = \dfrac{2 \times 2.50 \times 0.0576}{5}$
$I = \dfrac{2 \times 25\times10^{-1} \times 576\times10^{-4}}{5}$
$I = 2 \times 5 \times 576 \times10^{-5}$
$I = 576 \times10^{-4}$
$I = 0.0576\text{ kg}\cdot\text{m}^2$