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Equilibrium: Translational & Rotational

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  • 0:01 Equilibrium in Motion
  • 1:13 Translational Equilibrium
  • 4:32 Rotational Equilibrium
  • 7:11 Lesson Summary
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Lesson Transcript
Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

Learn about the requirements for an object to be under translational or rotational equilibrium. Then work through a couple example problems that show how we can use translational and rotational equilibrium to find the forces acting on an object.

Equilibrium in Motion

If you have ever suffered from an illness that caused dizziness, or even just stood up too fast after sitting for a long time, you might have heard someone tell you that you lost your equilibrium. When your body is in equilibrium, it is in a state of being physically balanced, and losing it causes you to feel dizzy. In physics, we also use the term equilibrium when talking about balance.

One common way equilibrium comes up is when looking at the motion of an object. However, we can have different types of motion and therefore different types of equilibrium. Two common types of motion are translational and rotational motion.

Translational motion happens when a body moves from one point to another. When you get up and travel from home to school or work, your body is experiencing translational motion since it is moving between two points. Rotational motion occurs when a rigid body revolves around an axis. Examples of rotational motion would be a ceiling fan turning or a wheel spinning. We'll look at both types of motion and learn about the states of equilibrium associated with them.

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.

force due to gravity equation

force due to gravity partially solved

force due to gravity solution

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.

horizontal forces sum

vertical forces sum

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.

finding frictional force part 1

finding frictional force part 2

finding frictional force solution

Next, we'll find the normal force.

finding normal force part 1

finding normal force part 2

finding normal force solution

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.

torque sum equation

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