Ever been curious why a book resting on a table doesn’t just float away or why a bridge doesn’t just crumble under all the weight of people, cars, and trucks on it? Well, we owe this all to equilibrium. In the field of physics, equilibrium occurs when the sum of all forces acting on an object all equate to zero—all forces are balanced.
In this article, we will take a look at the equilibrium of forces, how it is formed, the various types of forces, how it can be applied, and finally we shall take a look at some example problems and a step-by-step process to the solutions.
What is a force?
Before we can dip our toes into the equilibrium of forces, a prerequisite knowledge of “Force” is required.
So, what is a force? Simply put, a force is any and every push or pull that acts on an object. You just kicked a ball? That’s a force. You curled 50kg dumbbells a rep? Well, that’s impressive, but also a force. You just got slapped across your face? You’ve definitely felt a force. Think of a force as an invisible nudge.
Lets take a look at some common types of forces that you might encounter in your everyday life.
Gravitational Force: This is the pull force that all objects with mass have. On a normal bases, you and I can only feel the gravitational force of cosmic level objects. The earth for example, has a gravitational force that keeps us from floating away into space whilst also keeping the moon in its orbit. Other cosmic bodies like the sun and other planets also exert gravitational force. The magnitude of force all depends on the mass of the object.
Frictional Force: This is the force that occurs when two surfaces are in contact with one another—each trying to resist the sliding or rolling of the other. Different surfaces have varying levels of friction, that is why you would probably slip on the bathroom tiles if you were not careful but be totally fine when jogging by the sidewalk.
Tension Force: Tension is the pulling force experienced by objects when subjected to multidirectional forces in opposition, causing the object to stretch. Ever played a game of tug-of-war? The force experienced by the rope when being pulled from both directions is tension. Tension is also what causes stress and strain on materials.
Normal Force: When a book is placed on a table, a force prevents the book from going through the table. That force is what we call the “Normal Force”. The normal force prevents solid objects from passing through one another. In the book example above, the normal force would be acting against the weight of the book to stay in equilibrium.
The various forces above are but a few of the common types of forces that you might deal with in the day to day, there are many more. Forces are measured in Newtons (N).
Types of equilibrium of forces
Equilibrium in the context of the state of motion of an object is of two types. Static equilibrium and dynamic equilibrium.
Static equilibrium
Let’s go back to our example of the book on the table. Because the book is not moving and is stable in one place, we say that it is in static equilibrium. As you might have guessed, static equilibrium occurs when all the forces acting on an object cancel each other out leading to no movement.
Let’s paint a better picture. Take a block of solid iron which is placed on the ground. Now I come along with a forklift and start pushing this block with 100N of force, naturally the block would move in the direction I was pushing it. Now another co-worker of mine thought it would be funny, so he got another forklift and started pushing the block of iron in the opposite direction as me with an equal force of 100N. Due to the equilibrium of forces, the block would no longer move an inch because the forces cancel each other out. We now say that this block of iron is in static equilibrium.
Dynamic equilibrium
Dynamic equilibrium unlike static, occurs when an object is in motion. The motion is taking place at a constant speed and direction with no change in acceleration. An example of this might be a car moving at a constant 70km/hr with no change in its acceleration or deceleration. Even though the car is moving, we can say that it is in equilibrium.
Another example of dynamic equilibrium can be when a plane reaches cruising altitude. At this state, the plane is neither rising nor coming down. When it maintains a constant speed, we can extrapolate that it is in a state of dynamic equilibrium.
For an object to be in equilibrium (whether static or dynamic), two crucial conditions must be met:
- The summation on all forces acting on the object must be equal to zero \(\sum F = 0\). This ensures that there will be no change in motion.
- The summation of all torques (rotational force) acting on the object must also equate to zero \(\sum \tau = 0\).
Equilibrium can be further subcategorized into translational and rotational equilibrium. These are specific types of mechanical equilibrium that relate to the broader concepts of static and dynamic equilibrium.
- Translational equilibrium: This type of equilibrium occurs when the summation of forces acting on an object are balanced in all directions. This means that no matter how much force is applied, the object maintains the same position or moves with a constant velocity—at a state of static or dynamic equilibrium.
- Rotational equilibrium: This deals with forces that cause objects to rotate (torques). Let’s say you and a friend are sitting on a seesaw. When both of you are totally balanced—meaning no up or down movement, you are in rotational equilibrium. Essentially, the sum of all torques about a pivot point is zero.
Before we jump into solving problems, let’s talk about a handy dandy tool that you will be using to make your life much easier. The Free-Body Diagram is a very helpful tool that helps us visualize all the forces that are acting on an object. We use arrows to represent the direction and magnitude of each force.
Example problem 1
A 10 kg box is suspended from the ceiling by two ropes. One rope makes a 60-degree angle with the ceiling, and the other is horizontal. Find the tension in each rope.
Solution:
Step 1: Draw the free-body diagram showing the forces acting on the box.
Step 2: Break the tension in the angled rope into horizontal and vertical components using trigonometry.
- Vertical component: \( T_1 \sin(60^\circ) \)
- Horizontal component: \( T_1 \cos(60^\circ) \)
Step 3: Apply equilibrium equations.
- Vertically: \( T_1 \sin(60^\circ) = mg = 98 \, \text{N} \)
- Horizontally: \( T_1 \cos(60^\circ) = T_2 \)
Step 4: Solve for unknown tensions:
- First, use the vertical equation to find \( T_1 \):
\( T_1 = \frac{98}{\sin(60^\circ)} = 113.1 \, \text{N} \) - Then, solve for \( T_2 \) using the horizontal equation:
\( T_2 = T_1 \cos(60^\circ) = 113.1 \times 0.5 = 56.55 \, \text{N} \)
So, the tension in the angled rope is 113.1 N, and in the horizontal rope, it’s 56.55 N.
Example problem 2
A 4-meter-long beam is balanced on a fulcrum at its center. A 20 kg weight is hung 1 meter from the right end of the beam. Where should a 30 kg weight be placed on the left side to maintain equilibrium?
Solution:
Step 1: Draw the diagram showing the forces (weights) and the position of the fulcrum.
Step 2: Calculate the torques:
- Torque from the 20 kg weight on the right:
\( \tau_1 = 20 \times 9.8 \times 1 = 196 \, \text{Nm} \) - Torque from the 30 kg weight on the left:
\( \tau_2 = 30 \times 9.8 \times d \)
Step 3: Apply the equilibrium condition (\(\sum \tau = 0\)):
- \( \tau_1 = \tau_2 \), so \( 196 = 30 \times 9.8 \times d \)
Step 4: Solve for \( d \):
\( d = \frac{196}{30 \times 9.8} = 0.67 \, \text{m} \)
So, the 30 kg weight should be placed 0.67 meters from the center on the left side.
To wrap things up
Now that you know what equilibrium of forces is, how to draw free body diagrams, break down forces, and apply equilibrium equations, you are now set to go.
