How to double the time period of a simple pendulum For a simple pendulum operating at small amplitude, what change is required to double its period of oscillation?
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Adouble the mass of the bob
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Bquadruple the mass of the bob
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Cquadruple the length of the pendulum
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Ddouble the length of the pendulum
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Ehalve the local gravitational acceleration
Answer
Correct Answer: quadruple the length of the pendulum
Explanation
Introduction / Context:The period of a simple pendulum guides timing devices and dynamic isolation systems. Understanding parameter sensitivity helps engineers tune response without trial-and-error changes in mass or structure.
Given Data / Assumptions:
- Small-angle oscillations so that T = 2π * sqrt(l / g) applies.
- Mass of bob is concentrated; string is light and inextensible.
- Local gravity g is constant.
Concept / Approach:
Period T depends on length l and gravity g only. Mass does not appear in T. To change T by a factor, scale l accordingly using the square-root relationship.
Step-by-Step Solution:
T ∝ sqrt(l). To double T, set T_new / T_old = 2 = sqrt(l_new / l_old).Square both sides: 4 = l_new / l_old ⇒ l_new = 4 * l_old.Therefore, quadrupling the length doubles the period.Verification / Alternative check:
Plug into T = 2π * sqrt(l / g). If l → 4l, then T → 2π * sqrt(4l/g) = 2 * 2π * sqrt(l/g) = 2T, confirming the requirement.
Why Other Options Are Wrong:
(a) and (b) change mass, which does not affect T in the simple model. (d) doubling length increases T by sqrt(2), not 2. (e) while halving g would increase T by sqrt(2); it does not double T unless g is reduced by a factor of 4, which is unrealistic.
Common Pitfalls:
Believing mass influences period; forgetting the square-root dependence; applying the formula outside small-angle limits.
Final Answer:
quadruple the length of the pendulum