Find Dynamic Equilibrium: 5 Important Facts You Should Know

Physics’ equilibrium state differs slightly from that of chemistry. Equilibrium comes in two forms: static and dynamic. So, let us consider one of them: dynamic equilibrium.

A system or object is said to be in dynamic equilibrium when two or more processes happen simultaneously but have no net effect on one other. When there is no net influence of processes on each other, then the object moves in a constant motion, or more precisely, with zero acceleration and zero net force.

After knowing what exactly dynamic equilibrium is, let us now know how and when to find dynamic equilibrium with some solved problems in depth in this article.

When to find Dynamic equilibrium?

A system in dynamic equilibrium seems static at first, but a deeper, opposing process works to keep the micro-state stable. So, let us know when dynamic equilibrium is required.

When a particular object is in motion, it is important to find dynamic equilibrium. The dynamic equilibrium principle is used in the creation of lifting materials such as springs, ropes etc. Engineers use this concept to determine the forces acting on it while building other structures related to it.

How to find Dynamic equilibrium?

Our daily lives can benefit greatly from dynamic equilibrium. In light of this, let us know how to find dynamic equilibrium.

When real forces are considered, the D’Alembert’s principle is applied to calculate the dynamic equilibrium of any system. As per it, the difference between the net force acting on a system or body of mass particles and the time derivatives of the momenta is zero when it is projected onto any virtual displacement.

Strategy to find the Dynamic equilibrium

  1. Calculate the force applied to each particle of the body in Dynamic equilibrium.
  2. As the system is in a state of dynamic equilibrium, identify the force exerted on the body as a result of either its translational motion, rotational motion, or both.
  3. The equation that results provides the necessary conditions for a dynamic equilibrium system.

What is the formula for dynamic equilibrium?

The equation and principle of D’Alembert are employed to determine the dynamic equilibrium. So let us look at the formula used to determine the system’s dynamic equilibrium.

The following is a mathematical representation of D’Alembert’s principle:

i (Fi – miai) 𝛿ri =0

Where, 

i : The integral that is used to identify the variable in the system that corresponds to a specific particle

Fi : The total force that has been applied to the ith particle

mi : Mass of the ith particle

ai : Acceleration of the ith particle

𝛿ri : The virtual displacement of the ith particle

& miai : The time derivative representation.

Solved Dynamic equilibrium problems

1. What is the net amount of force acting on a car traveling at a constant speed of 70 km/h on a straight road?

Given: 

Speed of a car v = 70 km/h = 19.44 m/s

To Find: 

Net force acting on a car F = ?

Solution: 

As the speed of car is constant then acceleration of a car a = 0 m/s2

Thus, ma = 0 N ………..(1)

The D’Alembert’s principle is given by:

i (Fi – miai) 𝛿ri = 0 ……….(2)

Thus, 

(F – ma)  = 0

Putting the value of equation (1) in above equation we get net force:

F = 0

As a result, the sum of all the forces operating on the car is balanced and must be zero. It follows that the car is in dynamic equilibrium.

2. Find the condition for dynamic equilibrium for the block moving upward at a constant speed shown in the figure below.

Horizontally Moving Block In Dynamic Equilibrium

Solution:

As the block is in dynamic equilibrium the net force acting on a block must be zero. Thus,

Fnet = Fx + Fy = 0 ……….(1)

As no force acting on a block horizontally, the net force in the x direction is zero. Thus, we can write:

Fx = 0 ……….(2)

While in the y direction, the tension force T on the rope is in an upward direction, and the gravitational force mg due to its mass is in a downward direction. As a result, the net force acting in the y direction can be given as follows:

Fy = T – W = T – mg ……….(3)

Putting values of equation (2) and (3) in equation (1):

Fx + Fy = 0 +T – mg = 0

T = mg

If T > mg, the block would accelerate in the upward direction. If T < mg, the block would be decelerating and slowing down as it moved up, and eventually, it would stop.

Thus, T = mg is the required condition when a certain block is in dynamic equilibrium.

3. As seen in the figure below, when the box is dragged, it moves in the x direction at a constant velocity. Determine the conditions for dynamic equilibrium.

Dragged Box In Dynamic Equilibrium

Solution:

As the box is in dynamic equilibrium, net force acting on it would be zero. 

Net force in X direction:

Fx = T cos 𝜃 – fk = 0 ………..(1)

Where, fk is the kinetic friction force

Net force in Y direction:

Fy = N +T sin 𝜃 – fk = 0 ……….(2)

Thus, equation (1) and (2) are the required conditions for the box to attain dynamic equilibrium.

4. Determine the tension force of a block of mass 10 gm moving vertically in an upward direction at a constant velocity.

Given:

Mass of the block m = 10 gm = 0.01 kg

To Find: 

Tension force acting on a block T = ?

Solution:

As a result, the block is said to be in “dynamic equilibrium,” as it is moving with constant velocity. Thus, for dynamic equilibrium condition for the block is:

T = mg

Where g is gravitational acceleration (9.8 m/s2)

∴ T = 0.01 X 9.8

T = 0.098 N

Thus, the tension force of a given block is 0.098 N.

Conclusion:

We can conclude from this article that a system or body will have a constant velocity when it is in dynamic equilibrium since there will be no net force acting on it. Furthermore, D’Alembert’s principle is utilized to find dynamic equilibrium when dealing with real forces.

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