Static Equilibrium in Physics: Conditions, Forces, and Torques

What is Static Equilibrium?

Static equilibrium refers to a situation where an object is at rest or not moving, and all the forces acting on it are balanced. In other words, the object stays stationary because the forces trying to move it are equal and opposite.

Think of it like a table holding a book. The book doesn’t move because the force of gravity pulling it down is exactly balanced by the upward force from the table. So, the book is in static equilibrium.

Conditions for Static Equilibrium:

For an object to be in static equilibrium, two main conditions must be satisfied:

1. The sum of all forces acting on the object must be zero.
   • This means that the total force pulling in one direction must be exactly balanced by the total force pulling in the opposite direction.
2. The sum of all torques (moments) acting on the object must be zero.
   • Torque is the “turning force” that causes an object to rotate. For static equilibrium, the object should not rotate, so the total torque must also balance out to zero.

Let’s Break This Down with Some Examples:

1. Sum of Forces = 0

In static equilibrium, the forces acting on an object must cancel each other out. These forces can include things like gravity, friction, tension, and normal force.

For example, imagine a box resting on the floor:

• The force of gravity (weight) pulls the box down.
• The normal force from the floor pushes the box up.

For the box to be in static equilibrium, the downward force (gravity) must be equal to the upward force (normal force). Mathematically, this can be written as:

$$Sigma F = 0$$

Where:

• 
  $Sigma F$ 

  is the sum of all the forces acting on the object.

So, if the weight of the box is 50 N (Newtons), the normal force must also be 50 N.

2. Sum of Torques = 0

Torque (often called a moment or turning force) is what makes an object rotate. For an object to be in static equilibrium, it must not rotate, so the sum of all torques acting on it must be zero.

The torque depends on two things:

• The force applied to the object.
• The distance from the point where the force is applied to the axis of rotation (called the lever arm).

Imagine a seesaw with children sitting at different distances from the center. If one child is heavier or sitting farther from the center, the seesaw will tip. But if the torques are balanced, the seesaw will stay level.

Mathematically, the torque (

$tau$

) about a point is:

$$tau = F times d$$

Where:

• 
  $F$ 

  is the force applied.

• 
  $d$ 

  is the distance from the point of rotation (lever arm).

For an object in static equilibrium, the sum of all torques around any point must be zero:

$$Sigma tau = 0$$

Equations for Static Equilibrium:

To summarize the conditions for static equilibrium, we have:

1. Force Balance (No Movement in the Linear Direction):
   • For all forces acting on the object in any direction (up, down, left, right, forward, backward):

    

   $$Sigma F_x = 0 quad text{(Horizontal forces balance)}$$ 

   $$Sigma F_y = 0 quad text{(Vertical forces balance)}$$Where

   $F_x$is the sum of horizontal forces and

   $F_y$is the sum of vertical forces.

2. Torque Balance (No Rotation):
   • For all torques acting about any point, the total torque should be zero:

    

   $$Sigma tau = 0$$ 

Example: A Simple Beam in Static Equilibrium

Imagine a uniform beam that is supported by two points, one at each end, like a horizontal plank resting on two supports. Let’s say you place a heavy weight in the middle of the beam.

• The weight causes a downward force at the center of the beam.
• The two supports push up with equal forces (assuming the beam is uniform).

For this beam to be in static equilibrium, the following must hold:

1. The total vertical forces must be balanced:
   • The downward force from the weight equals the sum of the upward forces from the supports.

    

   $$F_1 + F_2 = W$$Where

   $F_1$and

   $F_2$are the forces from the two supports, and

   $W$is the weight of the object.

2. The total torque about any point must be zero:
   • If we choose the point where one of the supports is (say, at the left), then the torque from the weight about that point must be balanced by the torque from the other support.

    

   $$F_2 times L = W times frac{L}{2}$$Where

   $L$is the length of the beam, and the weight is at the center of the beam (half the length).

   Solving these equations will give you the values of the forces at the supports and show that the beam is in equilibrium.

Summary of Static Equilibrium

1. Static equilibrium means the object is at rest and not moving or rotating.
2. The sum of all forces acting on the object must be zero (
   $Sigma F = 0$ 

   ).

3. The sum of all torques (or moments) acting on the object must also be zero (
   $Sigma tau = 0$ 

   ).

4. These conditions ensure that the object stays in a stable, unmovable state.