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 m (a,a) (a,-a)
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- 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) NoneAnilAs seen in the image provided, a double-star system with stars of equal mass rotate in circular orbits around their mutual center of mass that is halfway between them. One of the stars (α) is bright. The other star (β) is its unseen dark companion. Our line of sight passes through the orbital plane such that once in every period, α approaches head-on, and once ever period it recedes directly away. The same is true for β. Suppose light always moves at speed c relative to the source that emits it (i.e., if v is the orbital speed of each star, light travels toward us at speed c + v from α when it is headed toward us, and at speed c - v when it is headed away from us, as depicted). The double-star system is a distance d away from Earth. How long would take light to get to Earth from α if the light is emitted when α is (i) coming toward us, and (ii) moving away from us?
- (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): = dxSuppose the 50 turn coil in the figure below lies in the plane of the page and originally has an area of 0.230 m². It is stretched to have no area in 0.100 s. X Bin XBin X V What is the direction? clockwise X counterclockwise X X What is the magnitude (in V) of the average value of the induced emf if the uniform magnetic field points into the page and has a strength of 1.75 T? X X 192 XExpress 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).
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