Translational Equilibrium
We say an object is in translational equilibrium when the sum of all the external forces acting on the object equals zero. Since a force is a mass times an acceleration, another way to look at this is that an object is in translational equilibrium when it is experiencing zero overall acceleration. This can mean either the object is not moving, or it is moving at a constant velocity.
If we can tell that an object is in translational equilibrium, we can use this to help find all the forces affecting that object. Imagine trying to push a 20-kilogram box along the floor. You push with an applied force of 200 Newtons, but the box doesn't budge an inch. Since our box isn't moving, it must be in translational equilibrium. Using the info given, let's find all the forces acting on the box.
In this example, there are actually four forces acting on the box: an applied force from you, a frictional force stopping you from moving the box, the force due to gravity pushing down, and the normal force from the floor pushing up on the box. The first thing we do is use our box's mass and the acceleration due to gravity to find the force due to gravity.
Now we just need to find the normal and frictional forces. To do this we are going to create two separate equations. In 2D we can split our forces up into those acting in the x direction (horizontally) and those acting in the y direction (vertically). The sum of the horizontal forces alone, and the sum of the vertical forces alone, both must equal zero since the box is in translational equilibrium.
Our force applied and the frictional force are the horizontal forces. The force due to gravity and the normal force are the vertical forces. With this we can find both our unknown forces. We'll start with finding the frictional force.
Next, we'll find the normal force.
The negative signs make sense because they show that the force due to gravity and the frictional force are acting in opposite directions from the normal force and applied force respectively.
Rotational Equilibrium
Rotational equilibrium works quite similarly to translational equilibrium. The main difference is that with rotation we are looking at torques instead of forces. So, much like translational equilibrium, we say an object is in rotational equilibrium when the sum of all the external torques acting on it equals zero. Again, we find that this must mean the object is either stationary or moving at a constant angular velocity. So, an object in rotational equilibrium must not be experiencing any angular acceleration.
When an object is in rotational equilibrium, we can use the fact that the sum of the torques must be zero to find the different individual forces acting on that object. One example for this is a beam balancing at its center on a fulcrum with two weights at either end. Each weight produces a torque on the beam that tries to rotate it around the fulcrum. The beam won't create any torque by itself as long as it is balanced with its center of mass on the fulcrum. So, the sum of the torques from weight 1 and weight 2 must equal zero.
Using what we just learned, let's try and find the force due to gravity created by weight 2 acting on the beam. Weight 1 creates a force due to gravity of 147 Newtons on the beam and is 2.4 meters from the fulcrum. Weight 2 is 1.3 meters from the fulcrum.
Diagram for example
 |
In the image, we can also see that torque 1 tries to rotate the beam counterclockwise, and torque 2 tries to rotate it clockwise. To show the direction of rotation, we give torques that rotate an object counterclockwise a positive sign and torques that rotate an object clockwise a negative sign.
Since we know that torque is a force times a distance, we have enough information to find the force due to gravity of weight 2.
Lesson Summary
Translational and rotational motion are two common types of motion in physics. Translational motion happens when a body moves from one point to another, and rotational motion occurs when a rigid body revolves around an axis. Both these types of motion have states of equilibrium associated with them.
An object is in translational equilibrium when the sum of all the external forces acting on the object equals zero. This also means an object is in translational equilibrium when it is experiencing zero overall acceleration. Therefore, it is either not moving or moving at a constant velocity.
Similarly to translational equilibrium, an object is in rotational equilibrium when the sum of all the external torques acting on it equals zero. In rotational equilibrium, an object either will not be moving or moving at a constant angular velocity. This must mean the object is experiencing zero angular acceleration.
Learning Outcomes
You should have the ability to do the following after this lesson:
- Define translational motion and rotational motion
- Recall when an object is in translational equilibrium and when an object is in rotational equilibrium
- Explain how to find the forces acting on an object in both translational and rotational equilibrium
