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A. A point mass m moves frictionlessly on a horizontal plane. An...
9. (13 points) A block of mass m is attached to a top of a spring...
9. (13 points) A block of mass m is attached to a top of a spring (spring constant k). The bottom of the spring is attached to a car of mass M that is free to move on a horizontal track. The spring is rigid enough that it is only able to move up and down, not side to side. See below picture. m Hlllllll car a. Write the Lagrangian in terms of x, y, x, and y. b. Write...
I think I have most of this question set, but would appractite step by step explaination...
I think I have most of this
question set, but would appractite step by step explaination of
questions e), f), g), and h). Thanks!
Two masses m1and m2 connected by a spring of elastic constant k slide on a frictionless inclined plane under the effect of gravity. Let a be the angle between the the x axis and the inclined plane, r the distance between the two masses, l the position of the first mass with respect to the top...
Solve the following problems: Problem 1: masses&springs Two masses mand m2 connected by a spring of...
Solve the following problems:
Problem 1: masses&springs Two masses mand m2 connected by a spring of elastic constant k slide on a frictionless inclined plane under the effect of gravity. Let a be the angle between the the x axis and the inclined plane, r the distance between the two masses, l the position of the first mass with respect to the top of the plane (see figure). Considering the top of the plane to be the zero for potential...
A block of mass m is attached to a top of a spring (spring constant k)....
A block of mass m is attached to a top of a spring (spring constant k). The bottom of the spring is attached to a car of mass M that is free to move on a horizontal track. The spring is rigid enough that it is only able to move up and down, not side to side. See below picture. Illlllll car a. Write the Lagrangian in terms of x, y, i, and y. b. Write the Hamiltonian in terms...
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
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....
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at ti...
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
A bead of mass m slides frictionlessly on a circle of wire with radius R. The...
A bead of mass m slides frictionlessly on a circle of wire with radius R. The circle stands up in a vertical plane and rotates about the z-axis with constant angular velocity . Write down the Lagrangian. Find the equations of motion. For an angular velocity greater than some critical angular velocity , the bead will experience small oscillations about some stable equilibrium point . Find and (). We were unable to transcribe this imageWe were unable to transcribe this...
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.
Question 3 3. Consider a plane pendulum consisting of a mass m suspended by a massless...
Question 3
3. Consider a plane pendulum consisting of a mass m suspended by a massless string of length I. Suppose that that time t-0 the pendulum is put into motion and the length of the string is shortened at a constant rate ot-a (ie. L(t)= Lo-at). Use the angle of the pendulum φ as your generalized coordinate. (a) (2 points) Obtain the Lagrangian and Hamiltonian for this system (b) (0.5 points) Is H conserved? How can you tell? (c)...
9. (13 points) A block of mass m is attached to a top of a spring (spring constant k). The bottom of the spring is attached to a car of mass M that is free to move on a horizontal track. The spring is rigid enough that it is only able to move up and down, not side to side. See below picture. m Hlllllll car a. Write the Lagrangian in terms of x, y, x, and y. b. Write...
I think I have most of this
question set, but would appractite step by step explaination of
questions e), f), g), and h). Thanks!
Two masses m1and m2 connected by a spring of elastic constant k slide on a frictionless inclined plane under the effect of gravity. Let a be the angle between the the x axis and the inclined plane, r the distance between the two masses, l the position of the first mass with respect to the top...
Solve the following problems:
Problem 1: masses&springs Two masses mand m2 connected by a spring of elastic constant k slide on a frictionless inclined plane under the effect of gravity. Let a be the angle between the the x axis and the inclined plane, r the distance between the two masses, l the position of the first mass with respect to the top of the plane (see figure). Considering the top of the plane to be the zero for potential...
A block of mass m is attached to a top of a spring (spring constant k). The bottom of the spring is attached to a car of mass M that is free to move on a horizontal track. The spring is rigid enough that it is only able to move up and down, not side to side. See below picture. Illlllll car a. Write the Lagrangian in terms of x, y, i, and y. b. Write the Hamiltonian in terms...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
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....
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
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.
Question 3
3. Consider a plane pendulum consisting of a mass m suspended by a massless string of length I. Suppose that that time t-0 the pendulum is put into motion and the length of the string is shortened at a constant rate ot-a (ie. L(t)= Lo-at). Use the angle of the pendulum φ as your generalized coordinate. (a) (2 points) Obtain the Lagrangian and Hamiltonian for this system (b) (0.5 points) Is H conserved? How can you tell? (c)...
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