Section 5.6: Equilibrium
Equilibrium occurs when the net force and net torque on an object are zero. Objects in equilibrium are either at rest or moving with constant velocity.
\[ \sum F_x = 0, \quad \sum F_y = 0, \quad \sum \tau = 0 \] where \( \tau \) = torque.
This applies to both translational and rotational equilibrium. Translational equilibrium ensures no linear acceleration, while rotational equilibrium ensures no angular acceleration.
Example 1: Beam in Equilibrium
A uniform 10 kg beam of length 5 m is supported at its ends. A 20 kg weight hangs 2 m from the left end. Find the reaction forces at the supports.
Let left support = \( F_L \), right support = \( F_R \).
Sum of vertical forces: \( F_L + F_R = 10*9.8 + 20*9.8 = 294 \, \text{N} \)
Taking torque about left support: \( 20*9.8*2 = F_R*5 \Rightarrow F_R = 78.4 \, \text{N} \)
Then \( F_L = 294 - 78.4 = 215.6 \, \text{N} \)
Practice Problems
- A uniform rod 4 m long and mass 12 kg is hinged at one end and held horizontally by a cable at the other. Find the tension in the cable.
- A seesaw of length 6 m has a child of 30 kg at one end and a child of 25 kg at the other. Find the pivot point where it balances.
- A sign of weight 200 N hangs from a horizontal beam of 5 m, supported at one end. Find the force at the support if the sign is at the far end.
- A uniform plank of mass 15 kg and length 3 m rests on two supports. A 10 kg weight is placed 1 m from the left support. Find the forces at each support.
- A uniform ladder of length 4 m rests against a frictionless wall. Find the forces at the base and the wall.