Questions -
Q1. A particle of mass m slides on a smooth inclined plane under the uniform gravitational field. Using d'Alembert's principle, obtain the equation of motion.
Q2. Consider the following spring-mass system on a smooth horizontal surface.
If x1 and x2 are measured from their (M1, M2) equilibrium positions, write down the tagrangian of the system and obtain equations of motion.
Q3. The point of a support a simple pendulum of mass m and length l connected to a spring of spring constant k and constrained to move in y-direction.
Using the covariant form of Newton's second law, d/dt (∂T/∂q·i) - ∂T/∂qi = Qi, where Qi = j=1∑n t→j(ext)· ∂r→/∂qi, obtain the equation of motion.
Q4. A mechanical system is described by the Lagrangian ⊥ = T - V with T = M/2[ω2 + α2θ· + 2αωθ·sin(θ-ωt)], V = mg(sinωt-αcosθ), where α, ω, m and g are constants.
a) Find an equivalent Lagrangian, Lew.
b) Obtain the energy function, h.
c) Is h a constant of motion. Explain.
d) Is h identical to the total energy (E) of the system. Explain.
e) Is E a constant of motion. Explain.