Consider a particle of mass m moving in a bound orbit with potential D- V(r) %3D ㅜ Using polar coordinates to find p, and pe. Is either one constant?
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- vative force, by explicitly showing that nds on only Q.n.4 Consider two particles of masses m1 and m2. Let ml be confined to move on a circle of radius a in the z = 0 plane, centered at x = y = 0. Let m2 be confined to move on a circle of radius b in the z = c plane, centered at x = y= 0. A light (massless) spring of spring constant k is attached between the two particles. a) Find the Lagrangian for the system. Q.n.5 Oral Viva(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): = dxConsider a thin disc of radius R and consisting of a material with constant mass density (per unit of area) g. Use cylindrical coordinates, with the z-axis perpendicular to the plane of the disc, and the origin at the disc's centre. We are going to calculate the gravitational potential, and the gravitational field, in points on the z-axis only. 1. Show that the gravitational potential 4(2) set up by that disc is given by p(2) = 2mGg | dr'; make sure to explain where the factor 27 comes from, and where the factor r' in the integrand comes from. 2. Evaluate this integral. 3. Approximate p(z), both for 0 R (i.e., for points very far away). You will need the following Taylor approximation: VI+x=1++O(x²), applied in different ways.
- Two particles, each of mass m, are connected by a light inflexible string of length l. The string passes through a small smooth hole in the centre of a smooth horizontal table, so that one particle is below the table and the other can move on the surface of the table. Take the origin of the (plane) polar coordinates to be the hole, and describe the height of the lower particle by the coordinate z, measured downwards from the table surface. Here, the total force acting on the mass which is on the table is -T r^ (r hat). Why?Explain this1. Consider a vertical plane in a constant gravitational field. Let the origin of a coordinate system be located at some point in this plane. A particle of mass m moves in the vertical plane under the influence of gravity and under the influence of an addition force f = -Ar"- directed toward the origin (r is the distance from the origin; A and a [#0 or 1] are constants). Choose appropriate generalized coordinates, and find the Lagrangian equations of motion (don't solve it). Is the angular momentum about the origin conserved? Explain.
- The circumference C of a circle is a function of its radius by C(r) = 2xr. Express the radius of a circle as a function of its circumference. Call this function r(C). r(C) = Preview Find r(187). r(187) = Interpret the meaning: O When the radius is 187, the circumference is r(187) O When the circumference is 187, the radius is r(187)Problem 3: A mass m is thrown from the origin att = 0 with initial three-momentum po in the y direction. If it is subject to a constant force F, in the x direction, find its velocity v as a function of t, and by integrating v find its trajectory. Check that in the non-relativistic limit the trajectory is the expected parabola. Hint: The relationship F = P is still true in relativistic mechanics, but now p = ymv instead of p = mv. To find the non-relativistic limit, treat c as a very large quantity and use the Taylor approximation (1+ x)" = 1 + nx when a is small.Consider two particles: p at the origin (0,0,0) = R³ with mass M > 0, and q at the point/position vector 7 = (x, y, z) = R³ with mass m > 0. Let G be the universal gravitational constant. (We will assume the MKS system of units.) The force F = F (7) felt by the particle q due to its gravitational interaction with particle p is: GMm 7(7)= == 7, for all 7 = (x, y, z) € R³\{0} . 17 Also consider the function ƒ : R³\{♂} → R given by GMm f(x, y, z) := TT , for all 7 = (x, y, z) € R³\{0} . Fix an arbitrary point/position vector = (x, y, z) in R³\{♂}. 2, calculate the (3) Calculat cade of the vector (4) Calculate the direction of the vector ₹(7). (5) Assume that is the total force on the particle q. Calculate the instantaneous acceleration, d, of the particle q when it is at the point 7 = (x, y, z).
- Evaluate the commutator è = [x², Pe** =?Let F = (z^2 cos y, −xz^2 sin y, 2xz cos y − cos z).a) Show that F is irrotational.b) Find a potential function f (x, yz) such that F = ∇f , and f (0, π, π/3) = 2Consider a particle of mass m moving in 1-dimension under a piecewise-constant po- tential. In region I, that corresponds to x 0. In region II, that corresponds to x > 0 the potential energy is V1(x) = 0. The particle is shot from = -∞ in the positive direction with energy E > Vo > 0. See the figure in the next page for a representation of V(x) as a function of x. Also shown in the graph (green dashed line) the energy E of the particle. (a) Which of the following functions corresponds to the wavefunction 1(x) in region I? (a1) Aeikiæ + Be-iki¤ ; (а2) Ае\1 + Bе-кӕ (a3) Aeikræ (а4) Ве- кта (b) Which of the following functions corresponds to the wavefunction 1(x) in region II? (b1) Сеkп* + De-ikr (62) C'e*I1* + De-*1¤