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.
Homework Answers
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Consider a mass point m that moves frictionless on the inner
side of a gutter (which...
der a frictionless ball of mass m rotating in a vertical circular cone as s the...
der a frictionless ball of mass m rotating in a vertical circular cone as s the following figure. The height of the oportionality constant k. Assume the gravitational constant is g Cons cone z is proportional to the radius r with Figure 1: A ball rotating in a frictionless cone a) Write the Lagrangian L in terms of variables r and the rotation angle ф. b.) Write the equations of motion from the Lagrangian. Do not solve them. c) Prove...
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...
1) Consider a block of mass M connected through the massless rigid rod to the massless...
1) Consider a block of mass M connected through the massless rigid rod to the massless circular track of radius a on a frictionless horizontal table (see the Figure). A particle of mass m is constrained to move on the vertical circular track. The distance between the center of the circular track and the center of mass of the block of mass M is constant and equal to L. Assume that there is no friction between the track and the...
(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
3) A particle of mass m is constrained to move on the inside surface of a...
3) A particle of mass m is constrained to move on the inside surface of a smooth cone of half- angle a. The particle is subject to a gravitational force. Determine a set of generalized coordinates and determine the constrains. Find Lagrange's equations of motion
2. 10 points Consider a body that is confined to move in a vertical plane, the...
2. 10 points Consider a body that is confined to move in a vertical plane, the x - z plane. The body has mass m and moves in the plane subject to the constant) gravitational force g in the – î direction) and an additional "central” force of the form f = - Ar-1/2, where y2 = 22 +22. This additional force is thus directed towards the origin. Choose the appropriate generalized coordinates and let the gravitational potential be zero...
Problem 4*: (Motion along a spiral) A particle of mass m moves in a gravitational field...
Problem 4*: (Motion along a spiral) A particle of mass m moves in a gravitational field along the spiral z = k0, r = constant, where k is a constant, and z is the vertical direction. Find the Hamiltonian H(z, p) for the particle motion. Find and solve Hamilton's equations of motion. Show in the limit r = 0, 2 = -g.
Two equal masses, m, are joined by a massless string of length L that passes through...
Two equal masses, m, are joined by a massless string of length L
that passes through a hole in a frictionless horizontal table.
First mass slides on table while the second hangs below the table
and moves up and down in a vertical line.
a.) Assuming the string remains taut, write down the Lagrangian
for the system in terms of the polar coordinates
of the mass on the table.
b.) Find the two Lagrangian equations of motion and interpret
the...
JUST ANSWER PART B A. A point mass m moves frictionlessly on a horizontal plane. An...
JUST ANSWER PART B
A. A point mass m moves frictionlessly on a horizontal plane. An unusual, anharmonic spring with unstretched length ro is attached between a pivot at the origin and the mass. Let the radial force exerted by the spring be given by Fr =-c(r-ro)" where c is a positive constant. Using plane polar coordinates r and θ: (i) Write down the Lagrangian L(r, θ,0) and use Lagrange's method to find the equations of motion for the mass...
Consider a simple pendulum of length / and mass m placed in a rail-road cart that...
Consider a simple pendulum of length / and mass m placed in a rail-road cart that has constant acceleration a in the positive x-direction. (Hint: This means that suspension point of the pendulum moves with acceleration a, this needs to be accounted for when considering motion of the pendulum) a) (11 pts.) Find the Lagrangian function of this pendulum. b) (11 pts.) Obtain Lagrange's equations of motion for this pendulum. c) (11 pts.) Find the Hamiltonian function of this pendulum....
der a frictionless ball of mass m rotating in a vertical circular cone as s the following figure. The height of the oportionality constant k. Assume the gravitational constant is g Cons cone z is proportional to the radius r with Figure 1: A ball rotating in a frictionless cone a) Write the Lagrangian L in terms of variables r and the rotation angle ф. b.) Write the equations of motion from the Lagrangian. Do not solve them. c) Prove...
1) Consider a block of mass M connected through the massless rigid rod to the massless circular track of radius a on a frictionless horizontal table (see the Figure). A particle of mass m is constrained to move on the vertical circular track. The distance between the center of the circular track and the center of mass of the block of mass M is constant and equal to L. Assume that there is no friction between the track and the...
(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
3) A particle of mass m is constrained to move on the inside surface of a smooth cone of half- angle a. The particle is subject to a gravitational force. Determine a set of generalized coordinates and determine the constrains. Find Lagrange's equations of motion
2. 10 points Consider a body that is confined to move in a vertical plane, the x - z plane. The body has mass m and moves in the plane subject to the constant) gravitational force g in the – î direction) and an additional "central” force of the form f = - Ar-1/2, where y2 = 22 +22. This additional force is thus directed towards the origin. Choose the appropriate generalized coordinates and let the gravitational potential be zero...
Problem 4*: (Motion along a spiral) A particle of mass m moves in a gravitational field along the spiral z = k0, r = constant, where k is a constant, and z is the vertical direction. Find the Hamiltonian H(z, p) for the particle motion. Find and solve Hamilton's equations of motion. Show in the limit r = 0, 2 = -g.
Two equal masses, m, are joined by a massless string of length L
that passes through a hole in a frictionless horizontal table.
First mass slides on table while the second hangs below the table
and moves up and down in a vertical line.
a.) Assuming the string remains taut, write down the Lagrangian
for the system in terms of the polar coordinates
of the mass on the table.
b.) Find the two Lagrangian equations of motion and interpret
the...
JUST ANSWER PART B
A. A point mass m moves frictionlessly on a horizontal plane. An unusual, anharmonic spring with unstretched length ro is attached between a pivot at the origin and the mass. Let the radial force exerted by the spring be given by Fr =-c(r-ro)" where c is a positive constant. Using plane polar coordinates r and θ: (i) Write down the Lagrangian L(r, θ,0) and use Lagrange's method to find the equations of motion for the mass...
Consider a simple pendulum of length / and mass m placed in a rail-road cart that has constant acceleration a in the positive x-direction. (Hint: This means that suspension point of the pendulum moves with acceleration a, this needs to be accounted for when considering motion of the pendulum) a) (11 pts.) Find the Lagrangian function of this pendulum. b) (11 pts.) Obtain Lagrange's equations of motion for this pendulum. c) (11 pts.) Find the Hamiltonian function of this pendulum....
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