The figure above shows a board placed on top of a fulcrum that is balanced by three masses (M₁, M₂, and M₃). If the mass of M₁ and M₂ are 60 kg and 30 kg, respectively, what is the mass of M₃?

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11ecba2d_1322_6b3e_a3bf_9b62af2e2341_MD0008_00 Torque is the tendency of an applied force to cause the rotation of an object around a pivot point located at some radial distance from where the force is applied. The magnitude of torque (τ) is expressed in terms of the magnitude of the applied force (F), the radial distance (r), and the angle (θ) between F and r: 11ecba2d_1322_6b3f_a3bf_3fffe1511db9_MD0008_00 τ = r⋅F sinθ Because torque increases as a force of equal magnitude is applied at greater distances from the pivot point, a mechanical advantage is observed in the context of rotational forces applied to a lever arm. However, when the sum of all torques acting on an object is equal to zero, an object is in rotational equilibrium (no rotation occurs): 11ecba2d_1322_6b40_a3bf_09a6bdd0a9c7_MD0008_00 ∑Torques=0 In this question, a board placed on a fulcrum is balanced by three separate masses. Each mass exerts a force on the board due to its weight. The resulting torque exerted on the left side of the fulcrum by the weight of M₁ is counterbalanced by the two torques exerted on the right side of the fulcrum by the weights of M₂ and M₃: 11ecba2d_1322_6b41_a3bf_e3f5cb1952c5_MD0008_00 Substituting the expressions for torque into the summation equation gives: 11ecba2d_1322_6b42_a3bf_613ec98c15a1_MD0008_00 Representing each torque as a product of weight and radial distance from the fulcrum yields: 11ecba2d_1322_6b43_a3bf_1d6af5b060ca_MD0008_00 Because the acceleration due to gravity g cancels from all terms and because all angles are equal to 90° (sin 90° = 1), the expression simplifies to: 11ecba2d_1322_9254_a3bf_d19e2e9d48d7_MD0008_00 11ecba2d_1322_9255_a3bf_a53386193195_MD0008_00 Using the values of M₁, M₂, r₁, and r₂ provided in the question yields: 11ecba2d_1322_9256_a3bf_293e4c739678_MD0008_00 11ecba2d_1322_9257_a3bf_01fed4b24a91_MD0008_00 11ecba2d_1322_9258_a3bf_55113e49a4d8_MD0008_00 11ecba2d_1322_b969_a3bf_7bbcc50683e4_MD0008_00 Therefore, the mass of M₃ must be approximately 17 kg for the board to remain in rotational equilibrium (Choice B). Educational objective: Torque is the tendency of an applied force to cause an object to rotate around a pivot point located at some radial distance from where the force is applied. Rotational equilibrium occurs when the sum of all torques acting on an object is equal to zero and results in no rotation.

The Figure Above Shows a Board Placed on Top of a Fulcrum