5.4: Forces in Translational Equilibrium
Objective
The student will:
- Understand how to apply Newton’s Second Law under equilibrium conditions in two dimensions.
Vocabulary
- static equilibrium: Objects in static equilibrium are motionless.
Introduction
There are many objects we do not want to see in motion. In the Figure above, the mountain climbers want their ropes to keep them from moving downward. We construct buildings and bridges to be as motionless as possible. We want the acceleration (and velocity) of these objects to be zero. For an object to be in static equilibrium (that is, motionless) the right-hand side of Newton’s Second Law,
But Joe won’t be hanging for very long, will he? You can see that the rope is slowly fraying against the mountainside (recall the original problem). Soon it will snap. But Joe’s in luck, because a rescue team has come to his aid. They arrive just in time to secure two more ropes to the mountain side and toss Joe the slack to tie around his waist before his rope snaps! Joe is saved. But does Joe thank the rescue team like any sane person would? No. Instead, still in midair, he pulls out a pad and pencil from his back pocket in order to analyze the forces acting on him (Figure below).
Static Equilibrium—Saving Joe
In order for Joe to remain in equilibrium, he must not move in the
The procedure for solving problems with forces in equilibrium is as follows:
- Place Figure above in a coordinate plane with the object at the origin.
- Resolve the tension vectors
T1 andT2 into theirx− andy− components. - Use Newton's Second Law:
∑Fx=0 and∑Fy=0 .
In order to solve this problem, we’ll need more information, including the angles that the ropes make with the vertical. The information is provided below, along with Figure below.
Find
The solution requires solving a set of simultaneous equations.
First, we find the components of vectors
Next we apply Newton's Second Law.
The first equation can be quickly simplified to give
Check your understanding
What general equation can be written for
Answer: The sum of the
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