A particle of mass m slides under the gravity without friction along the parabolic path y = a x² axis. Here a is a constant. The Lagrangian for this particle is given by-
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- A bead of mass m slides on a long straight wire which makes an angle a with the upward vertical and rotates with constant angular velocity o = w. Gravity g works vertically downwards. After choosing appropriate generalized coordinates, the Lagrangian is, mA) According to the Hartman Grobman theorem, the local behavior of the linearized system (saddle, node, etc.) is the same as that for the original system as long as the equilibrium is? B) For a two-dimensional system, the only non-hyperbolic case is a?Consider a vector Q that rotates at angular frequency ω about some fixed axis of rotation (~ω).Further, Q is at a fixed angle α relative to that axis. Show that the time rate of change of Q is given by
- Which of the following is the conserved quantity if the system having Lagrangian L= = m(x² + y²) – ² k(x² + y²). (a) Px (b) Py (c) L₂ (d) NoneThree identical cylinders of radius r are placed inside a hollow cylinder of radius R. All cylinder axes (perpendicular to the paper) are horizontal. There is no friction. The cylinders B and C are on the verge of separating (= infinitesimally separated, as shown). A B (a) From the statics equations for A and B, show that the angle between the normal under B and the vertical is given by tan 0 1 (The same result is obtained for A and C, of course, since B 3/3 and C have mirror image forces on them.) (b) By trigonometry of geometry, show that R must be r(1+2/7 Jin order for B and C to be on the verge of separating. (Find sin 0 and cos 0 from a triangle; don't find 0.) (Problem from a senior-year high school physics book used in England.) ogbo(a) For one-dimensional motion of a particle of mass m acted upon by a force F(x), obtain the formal solution to the trajectory x(t) in the inverse form: m = ₂√ 2 {E – V(x)} where V (x) is the potential energy and x0 is the position at t = 0. (b) If the force, F(x) is a constant then what is the equation of the particles trajectory (x vs t). t(x): = dx
- Obtain the inertia tensor of a system, consisting of four identical particles of mass m each, arranged on the vertices of a square of sides of length 2a, with the coordinates of the four particles given by (±a, ta, 0). Y m (-a,a) X (-a,-a). m O m (a,a) (a,-a)Express the Lagrangian for a free particle moving in a plane in a plane polar coordinates. From this proves that, in terms of radial and tangential components, the acceleration inpolar coordinates isa = (¨r − rθ˙2) er + (rθ¨ + 2 r˙ θ˙) eθ(where er and eθ are unit vectors in the positive radial and tangential directions).The inclination angle of a particle of mass m is adjustable, located on the moving edge. Inclined plane is horizontal at time t = 0 is in position. At t>0 moveable edge of the inclined plante is lifted by constant angular velocity of w to allow mass m to start to moving. Write down the Lagrangian equation of the mass m.
- Write down the inertia tensor for a square plate of side ? and mass ? for a coordinate system with origin at the center of the plate, the z-axis being normal to the plate, and the x- and y- axes parallel to the edges.A block of mass m = 240 kg rests against a spring with a spring constant of k = 550 N/m on an inclined plane which makes an angle of θ degrees with the horizontal. Assume the spring has been compressed a distance d from its neutral position. Refer to the figure. (a) Set your coordinates to have the x-axis along the surface of the plane, with up the plane as positive, and the y-axis normal to the plane, with out of the plane as positive. Enter an expression for the normal force, FN, that the plane exerts on the block (in the y-direction) in terms of defined quantities and g. (b) Denoting the coefficient of static friction by μs, write an expression for the sum of the forces in the x-direction just before the block begins to slide up the inclined plane. Use defined quantities and g in your expression. (c) Assuming the plane is frictionless, what will the angle of the plane be, in degrees, if the spring is compressed by gravity a distance 0.1 m? (d) Assuming θ = 45 degrees and the…Using Lagrangian formalism, solve the following problem: A solid cylinder is released from rest to roll without slipping down a ramp of slope ϕ. a) What is the best choice of generalized coordinates to be used?