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3) A particle of mass m is constrained to move on the inside surface of a...
4. A particle of mass m is constrained to slide without friction on the surface of...
4. A particle of mass m is constrained to slide without friction on the surface of a smooth circular bowl of mass M with inner radius R as shown in the figure. The bottom of the bowl lies on a horizontal table and is free to slide without friction along the table. All motion is constrained to the plane of the page. Assume uniform gravitati acceleration. =T-V- State the Lagrangian for this system. Derive the differential equations of motion for...
2.5.3 A particle is constrained to move on a vertical weightless circle of radius a in...
2.5.3 A particle is constrained to move on a vertical weightless circle of radius a in a uniform gravitational field. The circle rotates freely around a vertical diameter. Determine the equations of motion of the particle and derive the motion of the particle.
3. A particle of mass is constrained to move without friction along the x-axis, subject to...
3. A particle of mass is constrained to move without friction along the x-axis, subject to a potential energy siven by Ue) Uo/ constants. Show that for small oscillations about x 0, the particle undergoes simple harmonic motion. What condition on x is required for the oscillations to be "small" (i.e. simple harmonic)? Find the period Tof the oscillations. - 1) where Uo and b are positive
A particle of mass m is constrained to move along the x-axis and is subjected to...
A particle of mass m is constrained to move along the x-axis and
is subjected to a force given by
. Assuming the particle had an initial velocity of Vo and was at
the origin at t = 0, find an equation for the particle's velocity
and set up the integral from which the position equation as a
function of time could be determined. NOTE: You do not need to
evaluate the integral for the position as a function of...
Three spheres of equal mass m are constrained to move in one dimension along the line...
Three spheres of equal mass m are constrained to move in one dimension along the line connecting their centers. The three spheres are connected by three springs, as shown in the figure. The three springs have equal spring constants k In equilibrium, all three of the springs are at their respective natural lengths a. Choose a reasonable set of coordinates and find the equations of motion. b. Find the normal-mode frequencies.
Classical and Statistical Mechanics
A particle of mass ? slides along the inside of a smooth spherical bowl of radius ?. The particle
remains in a vertical plane (the ??-plane). (a) Assuming that the bowl does not move: (i) Write down the Lagrangian, taking the angle ? with respect to the vertical direction as
the generalized coordinate. (ii) Hence, derive the equation of motion for the particle. (b) Assuming that the bowl rests on a smooth horizontal table and has a mass ? and that the bowl can...
Consider a mass point m that moves frictionless on the inner side of a gutter (which...
Consider a mass point m that moves frictionless on the inner side of a gutter (which has a circular profile with radius R) and is subject to a homogeneous gravitational force field. There is an angle α between the gutter and the vertical direction. Write down the Lagrangian function (with suitable generalized coordinates) and obtain the equations of motion.
a) What is Schrodinger's equation for a particle of mass m that is constrained to move...
a) What is Schrodinger's equation for a particle of mass m that is constrained to move in a circle of radius R, so that psi depends only on phi? b) Solve this equation for psi and evaluate the normalization constant. (Hint: review the solution of Schrodinger's equation for the hydrogen atom) c) Find the possible energies of the particle. d)Find the possible angular momenta of the particle.
2) A particle of mass m, is attached to a massless rod of length L which...
2) A particle of mass m, is attached to a massless rod of length L which is pivoted at O and is free to rotate in the vertical plane as shown below. A bead of mass my is free to slide along the smooth rod under the action of a spring of stiffness k and unstretched length Lo. (a) Choose a complete and independent set of generalized coordinates. (b) Derive the governing equations of motion. m2
(10 pts) A stick with length l and mass M is allowed to fall due to...
(10 pts) A stick with length l and mass M is allowed to fall due to gravitational force. The surface on which the stick rests is frictionless. Use the angle φ and x position of point P shown in the figure as the two generalized coordinates and write down the equations of motion for the stick Sliding without friction
4. A particle of mass m is constrained to slide without friction on the surface of a smooth circular bowl of mass M with inner radius R as shown in the figure. The bottom of the bowl lies on a horizontal table and is free to slide without friction along the table. All motion is constrained to the plane of the page. Assume uniform gravitati acceleration. =T-V- State the Lagrangian for this system. Derive the differential equations of motion for...
2.5.3 A particle is constrained to move on a vertical weightless circle of radius a in a uniform gravitational field. The circle rotates freely around a vertical diameter. Determine the equations of motion of the particle and derive the motion of the particle.
3. A particle of mass is constrained to move without friction along the x-axis, subject to a potential energy siven by Ue) Uo/ constants. Show that for small oscillations about x 0, the particle undergoes simple harmonic motion. What condition on x is required for the oscillations to be "small" (i.e. simple harmonic)? Find the period Tof the oscillations. - 1) where Uo and b are positive
A particle of mass m is constrained to move along the x-axis and
is subjected to a force given by
. Assuming the particle had an initial velocity of Vo and was at
the origin at t = 0, find an equation for the particle's velocity
and set up the integral from which the position equation as a
function of time could be determined. NOTE: You do not need to
evaluate the integral for the position as a function of...
Three spheres of equal mass m are constrained to move in one dimension along the line connecting their centers. The three spheres are connected by three springs, as shown in the figure. The three springs have equal spring constants k In equilibrium, all three of the springs are at their respective natural lengths a. Choose a reasonable set of coordinates and find the equations of motion. b. Find the normal-mode frequencies.
2) A particle of mass m, is attached to a massless rod of length L which is pivoted at O and is free to rotate in the vertical plane as shown below. A bead of mass my is free to slide along the smooth rod under the action of a spring of stiffness k and unstretched length Lo. (a) Choose a complete and independent set of generalized coordinates. (b) Derive the governing equations of motion. m2
(10 pts) A stick with length l and mass M is allowed to fall due to gravitational force. The surface on which the stick rests is frictionless. Use the angle φ and x position of point P shown in the figure as the two generalized coordinates and write down the equations of motion for the stick Sliding without friction
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