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)?

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:

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.