A pirate captain has asked you to help him engineer a new plank to have prisoners walk overboard from. He needs the plan to fall overboard with the prisoners in order not to leave any evidence behind. He has a uniform beam of length 8m and a mass of 100kg is resting on two pivots. One pivot is at the left end, and the other is 5m from the left end. The prisoner will walk from the left end toward the right. Find where the prisoner will be when the plank tips. Assume the prisoner weighs 72kg.
To find where the prisoner will be when the plank tips, we need to determine the tipping point of the plank. The plank will tip when the total torque on one side of the pivot exceeds the total torque on the other side.
First, let's calculate the torques on each side of the pivot:
Torque = Force × Distance
On the left side of the pivot (with the prisoner):
Torque_left = Weight_of_prisoner × Distance_from_the_pivot
= mass_of_prisoner × gravitational_acceleration × Distance_from_the_pivot
On the right side of the pivot (without the prisoner):
Torque_right = Weight_of_plank × Distance_from_the_pivot
= mass_of_plank × gravitational_acceleration × Distance_from_the_pivot
Now, we need to set up an equation where the torques are balanced:
Torque_left = Torque_right
mass_of_prisoner × gravitational_acceleration × Distance_from_the_pivot = mass_of_plank × gravitational_acceleration × Distance_from_the_pivot
Since both sides have the same distance from the pivot, we can cancel out that term:
mass_of_prisoner × gravitational_acceleration = mass_of_plank × gravitational_acceleration
Substituting the given values:
72 kg × 9.8 m/s^2 = (mass_of_plank + 100 kg) × 9.8 m/s^2
Simplifying the equation:
705.6 = mass_of_plank + 980
mass_of_plank = 705.6 - 980
= -274.4 kg
Since the mass of the plank cannot be negative, the equation above has no real solution. This means that the plank will not tip.
Therefore, the prisoner will never reach a tipping point and will remain on the plank until it is either lowered or raised by other means.