**Subject :**Elements of Mechanical Engineering

## D' Alembert's principle

** D' Alembert's principle: **Newton's second law is

**F** = m**a**

We can write it

**F** (-m**a**) =0

We know that F is the resultant of external forces applied on the particle.

Considerring (*-m a*) as a force, we can say that the body is in equilibrium under the action of external forces and force (

*-m*). This fictitious force is known as inertial force, and the artificial state of equilibrium created is known as dynamic equilibrium. The apparent transformation of a problem in dynamics to one in statics has become known as D' Alembert's principle. D' Alembert's published his work in his "Traite de Dynamique" in 1743.

**a**Inertia force is a fictitiuous force. Assume that a particle is roatated in horizontal plane by means of a string.

For an external observer, the particle is moving and it has a centripetal acceleration *v ^{2}/r*. There is a tension

*T*which pulls the particle towards center. Newton's law can be applied and we get

ow, suppose the observer is sitting in the particle, itself. For him the particle is not moving, but he is seeing that the particle is being pulled by a force *T*. Thus he will feel that there is an outward force that is balancing the force. The fictitious outward force is called inertial force.

**One example:** A crate of mass *M* rests on a cart of mass *m*. The coefficient of friction between the crate and cart is and between cart and the road is . If the cart is to be pulled by a force *P*, such that crate do not slip, determine: (a) the maximum allowable magnitude of *P* and (b) the corresponding acceleration of the cart.

**Sol**:

Making free body diagram of mass *M*

Vertical force balance gives,

*N = Mg*

Frictional force, *F =**N*

*= Mg* This is the maximum possible acceleration without slip.

Now from the free body diagram of the crate cart,

*P-**(M m)g - (M m)a = 0 *

P = (M m)g (M m)g

*=(M m)g(** **)*