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4. The Lagrangian for the central force problem of a mass m traveling in polar coordinates...
A) Write the Lagrangian for a simple pendulum consisting of a point mass m suspended at...
A) Write the Lagrangian for a simple pendulum consisting of a point mass m suspended at the end of a massless string of length l. Derive the equation of motion from the Euler-Lagrange equation, and solve for the motion in the small angle approximation. B) Assume the massless string can stretch with a restoring force F = -k (r-r0), where r0 is the unstretched length. Write the new Lagrangian and find the equations of motion. C) Can you re-write the...
Compute the Euler-Lagrange equations for the Lagrangian: B8. where A, and V are arbitrary functions of the coordinates q. Find the conjugate momentum p, and show that the energy is Give the Hamil...
Compute the Euler-Lagrange equations for the Lagrangian: B8. where A, and V are arbitrary functions of the coordinates q. Find the conjugate momentum p, and show that the energy is Give the Hamiltonian. Show that wchere is a fuecion of q I a canonical trnsdormation Show that the com- bined transformation Ai = Ai + m-1 leaves the Hamiltonian invariant
Compute the Euler-Lagrange equations for the Lagrangian: B8. where A, and V are arbitrary functions of the coordinates q. Find...
Q 4. (a) A body of mass m is moving in two dimensions in a constant...
Q 4. (a) A body of mass m is moving in two dimensions in a constant z plane. Consider a coordinate system that rotates with constant angular speed 1 about the z-axis. In a fixed coordinate system (in the constant z plane), define the plane polar coordinates (r,0) while defining (r, ) as the corresponding plane polar coordinates in the rotating system. (i) In terms of the coordinate system rotating with constant angular speed 1, write down the kinetic energy...
A plane pendulum of length L and mass m is suspended from a block of mass...
A plane pendulum of length L and mass m is suspended from a
block of mass M. The block moves without friction and is
constrained to move horizontally only (i.e. along the x axis). You
may assume all motion is confined to the xy plane. At t = 0, both
masses are at rest, the block is at
, and the pendulum has angular deflection
with respect to the y axis.
a) Using
and
as generalized coordinates, find the Lagrangian...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
pls answer all the parts this is all the information 3. (a) Let L=L(x,y1, 41,99) where...
pls answer all the parts
this is all the information
3. (a) Let L=L(x,y1, 41,99) where = əy/ax, i = 1, 2, be a Lagrangian satisfying the Euler-Lagrange equation which is independent of y2. Show that al constant. aya You are given that the motion in the plane of a particle of mass m has La- grangian L = (1+2 +r202) – V(r), where r and 0 are polar coordinates, V is the potential and the dots indicate derivatives with...
Lagrangian Mechanics: A pendulum of mass m and length l hangs from the rear view mirror...
Lagrangian Mechanics: A pendulum of mass m and length l hangs from the rear view mirror in a car traveling with horizontal acceleration a. Assume the car starts from rest at time t=0. (Solve using Lagrangian Mechanics.) a) Draw a diagram of the situation. Write out the x and y coordinates of the position of the pendulum in the in terms of the angle of the pendulum, Φ, and the time t. b) Write out T, U, and L in terms...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force pr...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless ...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless spring, equilibrium length-Lo and spring constant- k). The other end of the spring is attached to the ceiling. Initially the spring is un-sketched but is making an angle θ° with the vertical, the mass is released from rest, see figure below. Let the instantaneous length of the spring be r. Let the acceleration due to gravity be g celing (a) (10...
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...
Compute the Euler-Lagrange equations for the Lagrangian: B8. where A, and V are arbitrary functions of the coordinates q. Find the conjugate momentum p, and show that the energy is Give the Hamiltonian. Show that wchere is a fuecion of q I a canonical trnsdormation Show that the com- bined transformation Ai = Ai + m-1 leaves the Hamiltonian invariant
Compute the Euler-Lagrange equations for the Lagrangian: B8. where A, and V are arbitrary functions of the coordinates q. Find...
Q 4. (a) A body of mass m is moving in two dimensions in a constant z plane. Consider a coordinate system that rotates with constant angular speed 1 about the z-axis. In a fixed coordinate system (in the constant z plane), define the plane polar coordinates (r,0) while defining (r, ) as the corresponding plane polar coordinates in the rotating system. (i) In terms of the coordinate system rotating with constant angular speed 1, write down the kinetic energy...
A plane pendulum of length L and mass m is suspended from a
block of mass M. The block moves without friction and is
constrained to move horizontally only (i.e. along the x axis). You
may assume all motion is confined to the xy plane. At t = 0, both
masses are at rest, the block is at
, and the pendulum has angular deflection
with respect to the y axis.
a) Using
and
as generalized coordinates, find the Lagrangian...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
pls answer all the parts
this is all the information
3. (a) Let L=L(x,y1, 41,99) where = əy/ax, i = 1, 2, be a Lagrangian satisfying the Euler-Lagrange equation which is independent of y2. Show that al constant. aya You are given that the motion in the plane of a particle of mass m has La- grangian L = (1+2 +r202) – V(r), where r and 0 are polar coordinates, V is the potential and the dots indicate derivatives with...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless spring, equilibrium length-Lo and spring constant- k). The other end of the spring is attached to the ceiling. Initially the spring is un-sketched but is making an angle θ° with the vertical, the mass is released from rest, see figure below. Let the instantaneous length of the spring be r. Let the acceleration due to gravity be g celing (a) (10...
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...
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