A particle is constrained to move on the circumference of a circle. If no external force is acting on the particle, show by using D’Alembert’s that the particle moves with uniform angular velocity.
Answers
Answer:
For a system of mass of particles, the sum of the difference of the force acting on the system and the time derivatives of the momenta is zero when projected onto any virtual displacement.
Explanation:
It is also known as the Lagrange-d’Alembert principle, named after the French mathematician and physicist Jean le Rond d’Alembert. It is an alternative form of Newton’s second law of motion. According to the 2nd law of motion, F = ma while it is represented as F – ma = 0 in D’Alembert’s law. So it can be said that the object is in equilibrium when a real force is acting on it. Here, F is the real force while -ma is the fictitious force known as inertial force.
Examples of D’Alembert Principle
1D motion of rigid body: T – W = ma or T = W + ma where T is tension force of wire, W is weight of sample model and ma is acceleration force.
The 2D motion of rigid body: For an object moving in an x-y plane the following is the mathematical representation: Fi= -mrc where Fi is the total force applied on the ith place, m mass of the body and rc is the position vector of the center of mass of the body.
This is D’Alembert’s principle.
Applications of D’Alembert’s Principle
D’Alembert’s principle is based on the principle of virtual work along with inertial forces. The following are the applications of D’Alembert’s principle:
Mass falling under gravity
Parallel axis theorem
Frictionless vertical hoop with a bead.
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