The elastic potential energy stored in a rubber band is related to its stretch, which is linked to the circumference of the cylinder it is placed around. The rubber band will stretch more around larger circumference cylinders, and thus store more potential energy when stretched.
The circumference \( C \) of a cylinder is given by the formula:
\[ C = \pi \times d \]
where \( d \) is the diameter of the cylinder.
Let's calculate the circumferences of each cylinder:
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For the 5 cm diameter cylinder: \[ C_1 = \pi \times 5 \approx 15.7 \text{ cm} \]
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For the 10 cm diameter cylinder: \[ C_2 = \pi \times 10 \approx 31.4 \text{ cm} \]
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For the 20 cm diameter cylinder: \[ C_3 = \pi \times 20 \approx 62.8 \text{ cm} \]
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For the 25 cm diameter cylinder: \[ C_4 = \pi \times 25 \approx 78.5 \text{ cm} \]
As the diameter increases, so does the circumference. Therefore, the rubber band around the cylinder with the largest diameter (25 cm) will have the largest circumference, which means it can store the greatest elastic potential energy when stretched.
Thus, the cylinder with the rubber band that has the greatest elastic potential energy is the one with the diameter of 25 cm.
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