Final answer:
By applying the principle of moments (torque) to the beam, with consideration of the beam's weight and position of the known masses, we solve for the unknown mass M that is required to maintain equilibrium.
Step-by-step explanation:
To find the value of the unknown mass M that keeps the beam in equilibrium, we can apply the principle of moments (also known as torque). The sum of clockwise moments around any pivot point must be equal to the sum of counterclockwise moments to achieve equilibrium. We'll choose point P as our pivot for simplicity.
The beam is 6.0 m long with supports 1.0 m from each end; the unknown mass M is situated at the left end and the known 10 kg mass at the right end. Since the beam itself has weight, we must consider its center of gravity, which is at its midpoint, 3.0 m from either end.
The moment due to the beam's weight is (4 kg)(9.8 m/s2)(2.0 m), using the midpoint of the beam as the center of mass and the distance from that midpoint to the pivot P (2.0 m). The moment due to the 10 kg mass is (10 kg)(9.8 m/s2)(5.0 m). Therefore, for equilibrium, M must provide a moment equal to the sum of the beam's moment and the 10 kg mass's moment, but in the opposite direction (counterclockwise).
Mathematically, this is set up as:
(4 kg)(9.8 m/s2)(2.0 m) + (10 kg)(9.8 m/s2)(5.0 m) = (M kg)(9.8 m/s2)(1.0 m).
Solving for M gives us the mass necessary to keep the beam in equilibrium.