Equilibrium

Question

A 115.00kg stack of physics texts is placed on a 6.00m long bench. What is the minimum reaction force each support leg must produce if the uniform beam itself has a mass of 10.0kg?

                              ______
      Texts books -->|______|
________________ |______|___________________________________
|__________________________________________________________|<--Bench
     leg -->| |                                              leg---->| |
________ |_|________________________________|_|_____________
|               |                 |                                         |                       |
0m           1m              2m                                      5m                   6m

Answer 1

Balance the torques:

Around the center of the table:

-> Text Book: Δx = -1m. F = -115g (downward force) => Torque: +115g Nm

-> Left Leg: Δx = -2m. F = F1 (upward force) => Torque = -2F1

-> Right Leg: Δx = 2m. F = F2 (upward force) => Torque = 2F2

Torque Balance: 0 = 115g + 2F2 - 2F1

 

Force Balance: the net force of both legs must support the table and books:

-> Books: F = -115g

-> Table: F = -10g

-> Left Leg: F = F1

-> Right Leg: F = F2

Force Balance: 0 = F1 + F2 - 125g

 

System of equations:

0 = 115g + 2F2 - 2F1

0 = -125g + F2 + F1

 

Multiply the second equation by 2

0 = 115g + 2F2 - 2F1

0 = -250g + 2F2 + 2F1

 

Add the equations together:

0 = -135g + 4F2

=> F2 = 33.75g = 330.75 Newtons (Right Leg)

 

Sub F2 into equation (1)

2F1 = 115g + 67.5g

F1 = 91.25g = 894.25 Newtons. (Left Leg)

 

*Intuition check: Since the books are more to the left of the table, you can expect the reaction force of the left leg to be larger (as shown). The relative magnitudes between the two legs depends on how much larger the books weight is compared to the table (roughly 10 times heavier, so the left leg should have a significant edge (not 10 times mind you) so 894 Newtons versus 330 Newtons seems reasonable.