Question

A window washer with a mass of 63.0 kg stands a distance, D = 0.600 m, from the left end of a plank of length, L= 2.10 m, with a mass of 19.0 kg. The plank is hung on two cables as shown. Find T₂, the tension in the right cable.

Answer 1

This problem involves statics: since there is no motion, the system is in equilibrium. It will be convenient to Newton's 2nd law in terms of rotation, which states that for a system in equilibrium, the sum of the torques about a pivot point is 0, or . It will be convenient to choose the left end where the left cable meets the plank to be the pivot point we're measuring stuff from. Recall that a torque is calculated using the equation , where r is the distance from the pivot point to where the force F is acting. Define positive torque to mean counterclockwise. All forces are acting perpendicular to the plank (r), so the torque reduces to .

Step 1) First notice that there are 4 forces in this problem. Let be the tension in the left cable that's holding up the plank on the left side. Let be the tension in the right cable that's holding up the plank on the right side. This is in fact what we want to solve for. Let the third force be from the window washer. He applies a force equal to his weight downward, . Let the fourth force be from the weight of the plank itself, equal to downward.

Step 2) Since the system is in equilibrium, the sum of the forces applied at various places on the plank results in a net 0 torque. We need to sum up the torques applied from each force and set it equal to 0. First though calculate the individual torques applied from each force. Force 1 is the tension in the left cable. Since the force is being applied at the pivot point itself, . Therefore the torque from the left cable is .

Step 3) Calculate the torque from the second force, the tension in the cable on the right. It is applied at a distance 2.10 m from the pivot point (the length of the plank). Therefore the torque from the tension in the right cable is .

Step 4) Calculate the torque from the third force, the window washer. He is 0.60 m from the pivot point, so the torque he applies to the pivot is . This is negative because it tends to rotate in the clockwise direction.

Step 5) Calculate the torque from the fourth force, the weight of the plank itself. The weight is applied at the center of mass of the plank, which is at the middle. The torque therefore is .

Step 6) Finally add up all the 4 torques and set it equal to 0, and solve for .

The tension in the right cable is 269.5 N.