Static balancing — a disturbing mass m₁ on a rotating shaft can be balanced by a single mass m₂ in the same plane at radius r₂ if which relation holds (r₁ and r₂ are radii of rotation)?
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Am1r2 = m2r1
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Bm1r1 = m2r2
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Cm1m2 = r1r2
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Dnone of these
Answer
Correct Answer: m1r1 = m2r2
Explanation
Introduction / Context: Static (force) balancing in a single plane requires the vector sum of centrifugal forces to be zero. When only one disturbing mass exists, a single balancing mass placed appropriately at the same plane can cancel it.
Given Data / Assumptions:
- Disturbing mass m₁ at radius r₁.
- Balancing mass m₂ at radius r₂ in the same plane.
- Both rotate at the same angular speed ω.
Concept / Approach: Centrifugal force F = m r ω². For zero net force, m₁ r₁ ω² and m₂ r₂ ω² must be equal in magnitude and 180° out of phase. Hence m₁ r₁ = m₂ r₂.
Step-by-Step Solution:
1) Write F₁=m₁r₁ω² and F₂=m₂r₂ω².2) Set F₁=F₂ for balance ⇒ m₁r₁ω²=m₂r₂ω².3) Cancel ω² (same plane and speed): m₁r₁=m₂r₂.Verification / Alternative check: Phasor diagram with two opposite vectors of equal length confirms zero resultant.
Why Other Options Are Wrong:m₁r₂=m₂r₁ — inverted; does not follow from force equality.m₁m₂=r₁r₂ — dimensionally inconsistent for balancing.none of these — incorrect because a correct relation exists.
Common Pitfalls: Forgetting the 180° angular placement; mixing up radii in the proportion.
Final Answer: m1r1 = m2r2.