Find the work done by the force field F = (2x + 4y, 4x + 8y) in moving a particle from (-1,1) to (1,1) along the path r(t) = (t,t2) in two ways:                     (a) by direct evaluation of the line integral;                                                         (b) by finding a scalar potential and using the gradient theorem.

Q: Evaluate the volume between the surface z = 3 + 2x + xy, and the rectangle in the xy plane given by…

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Q: U(x) = U, e-x/a A particle of mass m moves in the one-dimensional potential x2 U(x) = U,

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Q: A particle moves in one-dimensional motion through a field for which the potential energy of the…

A: given quantities: U(x)=(A)/(x3) -B/x A=0.15 B=3.68 (a) To find the equilibrium position , first ,…

Q: Consider a particle of mass m moving in a bound orbit with potential D- V(r) %3D ㅜ Using polar…

A: Radial momentumPr and angular momentumPθ and to determine whether either one is constant.

Q: F(x, y) = (5y-x²)i + (5x + y)j conservative not conservative If it is, find a potential function for…

A: Given F(x,y)= (5y-x2)i +(5x+y)therefore , for the force to be conservative , curlF = 0 .

Q: Let charge Q be uniformly distributed on a circular ring defined by a <p<b and shown in Figure 4.26.…

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Q: Calculate the energy, corrected to first order, of a harmonic oscillator with potential 1 V(x) =kx +…

A: Given, Vx=12kx2+ωx4+ω2x6The 1st excited state wave function for harmonic oscillator potential is…

Q: A particle is moved by the force field F(x, y) = [e-r²y, ev+ry²] from (-1, 1) to (1,1) along the…

A: We will answer the question by using formula for work done by a force. The detailed steps are given…

Q: For the scalar potential field f = x^2*y^3 + x*y^2*z^3 + c, what is the direction and magnitude of…

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Q: Consider the points A(−4, 2, 0), B(7,3,−2) and C(−2, −3,1) (a) Find a vector of length √6 in the…

A: Given points A(-4,2,0), B(7,3,-2) and C(-2,-3,1) AB→=7-(-4)i^+3-2j^+-2+0k^AB→=11i^+1j^-2k^…

Q: Show that the force vector is conservative, hence Find the potential function

A: A vector-is said to be conservative if it’s curl comes out to be  zero and then only corresponding…

Q: Calculate the work done in moving a 4-C from B(1,0,0) to A(0,2,0) along a path on the plane z = 0 in…

A: Given: The magnitude of charge is 4 C. The electric field is 5xax+5yay Vm.Introduction: Electric…

Q: We consider a particle of mass m that has a (dimensionless) position ¤(t) E R on a one-dimensional…

A: Disclaimer: Above is an example of a multiple-part question. It is not indicated which part to be…

Q: particle moves in a central force field given by the potential ar V = -k- re k and a are positive…

A: Given:- 2.) A particle moves in a central force field given by the potential  V = -Ke-arr where k…

Q: Show that the line integral -Q E•AL is not dependent on the path selected between B to A but only…

A: A point charge Q is situated at the origin. Then electric field at a distance r from the center O…

Q: 2-a) Using the matrix treatment derive the energy eigen values for the Hint: harmonic oscillator…

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Q: Prove that the vector field F(x,y,z) = (x^2 + yz)i − 2y(x + z)j + (xy + z^2)k is incompressible, and…

A: We have been given a vector field and we need to show that its incompressible and also need to find…

Q: A particle of mass m moves in the one-dimensional potential U(x) = U, Se-x/a Sketch U(x). Identify…

A: Here, we use the maxima/minima conditions to get the required.

Q: commutator

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Q: Consider the velocity field, V = (x − 2y)i — (2x + y)j. What is the value of the velocity potential…

A: Velocity potential ϕ(x,y,z) and velocity v→ are related by  v→=∇ϕ(x,y,z) = ∂ϕ∂xx^ + ∂ϕ∂yy^ + ∂ϕ∂zz^

Q: Show that the force F = -k(yz, rz, ry), where k is a constant, is conservative and determine the…

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Q: A particle moves in a potential given by U(x) = -7 x5 + 52 x + 85  (J).  Calculate the x coordinate…

A: The given potential is Ux=-7x5+52x+85                                                   1 At…

Q: Consider a sphere of radius R, the potential at the surface of the sphere is given by: V (R, 8) =…

A: Consider a sphere of radius R, the potential at the surface of the sphere is given by: V(R,theta) =…

Q: Show that a covariant definition of the energy (valid in any reference frame) is E = -P#Uµ. (This…

A: Uμ  =  Four velocity  =  dxμdτ   a contra varient tensor of rank 1  and we know , dτ = dtγ   Now Uμ…

Q: Determine the area of the triangle having vertices at the three points (3, 4, 5), (4, 5, 3) and (5,…

A: Suppose the three points A(3,4,5), B(4,5,3) and C(5,3,4) forming the following vectors in any…

Q: Consider the Lennard-Jones potential between atoms in a solid material. Find the dependence of the…

A: The Lennard-Jones potential is given by, V=4Eσx12-σx6 where E is the depth of the potential well or…

Q: Potential of a Modified Infinite Plane Consider an infinite plane at z = 0 where the potential is…

A: The potential along the axis of the circle s=0, is given by, Vx→=Vo1-za2+z2.

Q: 2.2 Find the potential associated with F=(2xy+z')i+x*j+3xz'k. 2.3 Find the work done in moving a…

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Q: A sphere of radius R centered at the origin carries charge density ρ(r, θ) = k*(R/r^ 2) (R − 2r) sin…

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Q: A particle of mass m moves in the logarithmic potential V (r) = C In (r/ro). Show that: (a) All…

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Q: 2. f= (2y + + x j +3yz %3D

A: Given, F=2xy+z3i→+x2j→+3yz2k→∇×F=i^j^k^∂ ∂x∂ ∂y∂ ∂z2xy+z3x23yz2Solving we get=∂ ∂y3yz2-∂ ∂zx2i^-∂…

Q: Calculate the commutator 2m between the kinetic energy and the position operators for a particle…

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Q: Verify that each of the following force fields is conservative. Then find, for each, a scalar…

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Q: prove this force are conserved F(x,y)=2a(x-y)i-a(2x-y)j Where a is constant

A: Given that:- Force F(x,y)=2a(x-y)i^ - a(2x-y)j^

Q: (a) For one-dimensional motion of a particle of mass m acted upon by a force F(x), obtain the formal…

A: We need to show the given relation for t as a function of x.

Q: Consider a thin disc of radius R and consisting of a material with constant mass density (per unit…

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Q: Consider in three-dimensional space a particle that moves at a central potential . Then the equation…

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