D'Alembert's Principle

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Introduction to D'Alembert's Principle

D’Alembert principles state that the addition of difference of the force that acts on the system and the time derivatives of the momentum is zero.

This statement is correct only when the force and time derivatives are projected onto any virtual displacement. 

Also, the principal has another name called Lagrange- d’Alembert principle. The name of this principle is based on the french mathematician and physicist Jean le Rond d’Alembert.

We can say that this law works as an alternative for Newton’s second law of motion.


D’Alembert law Mathematical Representation

D'Alembert theorem is another form of newton’s second law. In newton’s second law, it is said that $F = ma$.

Well, in the case of d Alembert law in mechanics, it is represented as $F – ma = 0$.

As per the above expression, a statement is derived. According to that statement, we can say when a body can get into its equilibrium state when there is an external force acting on it.

(Image to be added soon)

(Portrait of Jean le Rond d’Alembert)

The d Alembert principle derives $F$ as the real force and $‘-ma’$ as the fictitious force, also known as inertia force. 

The mathematical expression of d’Alembert’s Principle is given in the following way:

$\sum_{i}\left(F_i-m_ia_i\right)\delta r_i = 0$ 


The Terms Used in the Above Expressions Are

  1. $i =$ the integral 

    1. This is used for the identification of variables. It is also corresponding to the specific particle in the system.

  2. $m_i =$ the mass of the ith particles

  3. $F_i =$ Total applied force on the $i^{th}$ place

  4. $\delta r_i =$ the virtual displacement of $i^{th}$ particle

  5. $a_i =$ the acceleration of $i^{th}$ particles

  6. $m_ia_i =$ the time derivative representation


State D'Alembert's Principle

Students can put forward the relation of virtual work by using D’Alembert’s principle. The result of the expression will be zero. 

The derivation is given below:

For total force on each particle:  $F_{i}^{(T)} = m_ia_i$

$F_{i}^{(T)} - m_ia_i = 0$

This is the inertial force so that force is moved to the left. The above expression is also mentioning quasi-static equilibrium.

By equating it with the virtual work, the expression will be

$\delta W=\sum_i F_{i}^{(T)} \cdot \delta r_i - \sum_i m_ia_i \cdot \delta r_i = 0$

When you separate the applied force and constraint force, the expression will be:

$\delta W = \sum_i F_i \cdot \delta r_i + \sum_i C_i \cdot \delta r_i - \sum_i m_ia_i \cdot \delta r_i = 0$ 

So, the final expression for the D’Alembert’s principle is:

$\delta W = \sum_i \left( F_i - m_ia_i\right) \cdot \delta r_i = 0$ 


Examples of D'Alembert's Law

As we know, d’Alembert’s principle is the derived form of Newton’s second law. The mathematical approach done by D’ Alembert was quite beneficial for the use of physics.

We use its application in some significant sectors such as:

  1. The 1D Motion of a Rigid Body

In this application, tension is the term we use frequently here. 

The mathematical expression is, $T – W = ma $

Or, $T = W + ma$.

Here, 

  • $T =$ tension force of the wire

  • $W =$ weight of the sample model

  • $ma =$ Acceleration force


  1. The 2D Motion of a Rigid Body

In this application, when an object is in motion on an x-y plane, then the mathematical expression for their motion is,

$F_i = - m \times r_c$

Here, 

  • $F_i =$ The total force applied on the $i^{th}$ plane

  • $r_c =$ position vector of the centre of mass of the body

  • $m =$ mass of the body


Application of D’Alembert’s Principle

D’Alembert’s Principle is completely dependent on the working principle of virtual work. It deals with the inertial forces. 


Here Are Some of the Crucial Applications of D’Alembert’s Principle:

  • Mass under the gravitational force

  • Frictionless Vertical Hoop with a bead

  • Parallel axis theorem 


Do You Know the Second Law of Newton?

According to Newton, the second state explains that the force is equal to the rate of change of momentum. This is applicable for a body having a constant mass. The force acting on it is equal to the product of mass and acceleration of it.

As per the law, there are two types of variables that create an impact on the acceleration of an object. They are:

  • The net force acting on the object 

  • The mass of the object

The body’s acceleration is directly proportional to the total force. This is the force that acts on the body due to some external facts. Also, the acceleration is inversely proportional to the mass of the body. 

From the above statements, we conclude that the force that acts upon an object is raising due to the increase of the acceleration of the object. 

Also, we notice the increase in the mass of an object when the acceleration of the same object is decreased.


Application of Newton’s Second Law

The application of Newton's second law signifies different types of force that enable the movement in an object or acts as a stopping force. 

  • Pushing a cart

  • Kicking the ball

  • Dragging a chair or table

  • Moving of brick

  • The motion of a car

  • Walking of two people and so on.

FAQs (Frequently Asked Questions)

1. Calculate the acceleration of the block having a mass 5 kg when two forces are acting upon it from two directions. The figure is given below that shows the forces that act upon the block.

(Image to be added soon)

Data given, mass of the block = 5 kg

Force from the left side = 20 N

According to figure, Force from the right side = 30 N

So, the total force is Fnet = 30 – 20 = 10 N

We know the formula of force i.e. F = ma 

So, Acceleration a = F/m = 10/5 = 2 kg

2. How do you define the principle of virtual work?

The principle that stands for virtual work is defined as the virtual work done by the forces on an object when that object is under an equilibrium state. The forces applied to the object will be zero. This explanation is very much similar to Newton’s laws.

3. Where do we use D’Alembert’s principle?

D’Alembert’s principle is widely used in the analysis of dynamic problems. They are helpful to learn those problems by reducing them into static equilibrium problems.

4. Mention Two everyday life examples associated with newton’s second law.

Here are the examples associated with newton’s second law

  1. Pushing a car

  2. Acceleration of golf ball in a golf game

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