What mass would you hang on the right side of the system in the figure to balance it -- that is, to make the clockwise and counterclockwise torques equal? (Let x1 = 30 cm and x2 = 12 cm.)
To balance the system, we need to make sure that the clockwise and counterclockwise torques are equal. The torque is calculated by multiplying the force applied to an object by the distance from the fulcrum (or pivot) point. In this case, the fulcrum point is at the center, and we have two torques acting in opposite directions.
First, let's determine the torques acting on the system. The counterclockwise torque is produced by the 10 kg mass, and its distance from the fulcrum is x1 = 30 cm. The clockwise torque is produced by the unknown mass we want to find and its distance from the fulcrum is x2 = 12 cm.
The counterclockwise torque can be calculated as follows:
Torque_counterclockwise = force_counterclockwise * distance_counterclockwise
The clockwise torque can be calculated as follows:
Torque_clockwise = force_clockwise * distance_clockwise
Since the system is in equilibrium, the torques must be equal:
Torque_counterclockwise = Torque_clockwise
Now, let's equate the torques and solve for the unknown force (mass). Using the equation above and assuming the force due to gravity as g = 9.8 m/s^2:
force_counterclockwise * x1 = force_clockwise * x2
Replacing the forces with their corresponding masses:
(mass_counterclockwise * g) * x1 = (mass_clockwise * g) * x2
Rearranging the equation to solve for the unknown mass (mass_clockwise):
mass_clockwise = (mass_counterclockwise * x1) / x2
Plugging in the values: mass_counterclockwise = 10 kg, x1 = 30 cm (0.3 m), and x2 = 12 cm (0.12 m):
mass_clockwise = (10 kg * 0.3 m) / 0.12 m
mass_clockwise = 75 kg
Therefore, to balance the system, you would need to hang a mass of 75 kg on the right side of the system.