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A particle moving in one dimension interacts with a potential V(x).In a stationary state of this system, Show that:
1/2<x partial V/Partial x>=<T>
Where T=P2/2m is the kinetic energy of the particle.
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- In Lagrangian mechanics, the Lagrangian technique tells us that when dealing with particles or rigid bodies that can be treated as particles, the Lagrangian can be defined as: L = T-V where T is the kinetic energy of the particle, and V the potential energy of the particle. It is also advised to start with Cartesian coordinates when expressing the kinetic energy and potential energy components of the Lagrangian e.g. T = m (x² + y² + 2²). To express the kinetic energy and potential energy in some other coordinate system requires a set of transformation equations. 3.1 Taking into consideration the information given above, show that the Lagrangian for a pendulum of length 1, mass m, free to with angular displacement - i.e. angle between the string and the perpendicular is given by: 3.2 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.2 L = T-V = ²² +mg | Cos Write down the Lagrange equation for a single generalised coordinate q. State name the number of generalised coordinates in problem 3.1. Hence write…m, A particle of mass 2.00 × 10-10 kg is confined in a hollow cubical three-dimensional box, each edge of which has a length, 2.00 × 10-10 and for which the potential energy function is zero inside, and infinite outside, the box. The total energy of the particle is 2.47 × 10-37 J. FindA 8 kg particle is in a potential given by U(x) = 1 x^4 - 7 x^2 - 3 x + 8 (J). Calculate the acceleration of the particle when it is at x = 1 m, in m/s2. (Please answer to the fourth decimal place - i.e 14.3225)
- Answer q1a Q=6A new type of force was discovered by physicists with the following expression: where alpha & beta are constants, and x is the position. The expression above was obtained from the interaction of a massless Higgs Boson (a type of particle) and a black hole. Quantum physicists then decides to design and build a machine that is able to move the Higgs Boson from x2 to x1. How much work should the machine do to achieve this feat? (For simplicity, consider that no energy is lost in the process) Solution To determine the work done we apply the following W = dx Evaluating the above, we get W = | | + e + x for the limits from xi to xf substituting x1 and x2 as the limits, the work done is expressed as W = | / | + ( x1 - ) + ( x15 - x25 ) Kindly box the answers so that I won't be confused, Thank you!!Consider a force in R3 defined by f(x,y,z)=(6*x*y^3, 9*x^2*y^2+2*y,0). Find the potential energy U(x,y,z) such that U(0,0,0)=0.
- Show that the work done by the force field F=(y' – 2xyz")î+(3+2.xy – x*z')}+(6z° – 3x°yz*)& is path independent and calculate the work done by this force field in moving a particle from (1, -1, 2) to (2,3,1).Energy and potential Prove that F = (y, -x, 0) is not conservative.Consider the 3-dimensional force field ⃗ F = (x^2 − ze^y)⃗i + (y^3 − xze^y)⃗j + (z^4 − xe^y)⃗k:(a) Show that ⃗ F is conservative.(b) If ⃗ F is conservative, find the corresponding potential function f (x, y).(c) If an object travels on a path ⃗r (t), (t_0 < t < t_f ), does the work done bythe force field depend on the path taken? Find the work done on the object movingon the path ⃗r (t) by the force field ⃗ F , if ⃗r (t_0) = (3, 1, 2) and ⃗r (t_f ) = (6, 2, 5)