State D'Alembert's principle and mention its applications in Plane Motion.
Answers
Answer:
The D’Alembert’s, also known as Lagrange–d ‘Alembert principle states that the sum of the incremental virtual works done by all external forces "Fi" acting in conjunction with virtual displacements "δsi" of the point on which the associated force is acting is zero.
δW = ∑i Fi . δsi = 0 …………………(i)
Explanation:
This technique is useful for solving statics problems, with static forces of constraint. A static force of constraint is one that does no work on the system of interest, but merely holds a certain part of the system in place. In a statics problem there are no accelerations. We can extend the principle of virtual work to dynamics problems, i.e., ones in which real motions and accelerations occur, by introducing the concept of inertial forces. For each parcel of matter in the system with mass m, Newton’s second law states that
F = ma…………………………(ii)
We can make this dynamics problem look like a statics problem by defining an inertial force
Fⁿ = −ma …………………………(iii)
Rewriting equation (ii) as
Ftotal = F + Fⁿ = 0
D’Alembert’s principle is just the principle of virtual work with the inertial forces added to the list of forces that do work:
δW = ∑iFi · δsi + = ∑iFjⁿ · δsj = 0
Applications of D’ Alembert’s principle:
1. Mass falling under gravity:
- An example would be a mass “m” falling under the effect of a constant gravitational field “g”. With “z” positive upward, the force on the mass is “−mg” and the work due to this force under vertical displacement “δz” is: δWG = −mgδz. The inertial force is “−m (d² z/dt²)” and the work is: δWI = −m (d²z/dt²) δz. Setting the sum of the two to zero gives us
−mgδz − m (d² z/dt²) δz = − mg + m (d² z/dt²) δz = 0
From which we infer the expected result
d² z/dt²= −g
2. Bead on frictionless vertical hoop:
- The force of gravity is not in the direction of motion. The component of gravity normal to the hoop does no work on the bead. Nor does the force of the hoop on the bead that constrains the bead to move in a circle. The work on the bead due to gravity for a small displacement δs = Rδφ along the wire is δWG = −mgR sinφ δφ. The acceleration of the bead also has two components, a radial component ar = −v²/R, where the tangential velocity is v = ds/dt = R(dφ/dt), and a tangential component at = d² s/dt² = R (d²φ/dt² ). The radial component of the inertial force mv²/R does no work. However, the tangential component −mat = −mR(d²φ/dt²) does: δWI = −mR (d2φ/dt2) δs = −mR2(d2φ/dt2) δφ. D’Alembert’s principle thus gives us
−mgR sin φ − mR2 (d2φ/dt2)δφ = 0
3. Atwood’s machine:
- The state of the Atwood’s machine is determined by the positions of the two masses along the U-shaped coordinate s looping over the frictionless pulley with the string. The work done by gravity on the left-hand mass under the displacement “δs” is
δWL = −mgδs,
while the gravity acting on the right-hand mass produces work
δWR = +Mgδs.
The inertial work on the two masses gives
δWI = −(M + m) (d2 s/dt2) δs,
resulting in δW = (M − m) g − (M + m) (d2 s/dt2) δs = 0
or d2 s/dt2= g M − m M + m
Note that the force of constraint by the pulley (which is assumed to be free to rotate but held rigidly in place and with negligible moment of inertia) does not enter the problem at all.
