Concept:Centripetal force depends directly on mass and inversely on the radius. We track these changes by plugging the new variables into the formula.
Formula:$$ F_c = \frac{mv^2}{r} $$
Solution:- Let the initial force be \( F = \frac{mv^2}{r} \).
- The new mass is \( \frac{m}{2} \) and the new radius is \( 2r \).
- Substitute these into the formula: \( F' = \frac{(\frac{m}{2})v^2}{2r} \).
- Simplify the fraction: \( F' = \frac{mv^2}{2 \times 2r} = \frac{mv^2}{4r} \).
- Extract the original formula: \( F' = \frac{1}{4} \left( \frac{mv^2}{r} \right) = \frac{1}{4}F \).
- Therefore, the force becomes one-fourth.
Why other options are incorrect:- Option D happens if you mistakenly think halving mass and doubling radius cancel each other out (they compound, not cancel, since radius is in the denominator).
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