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3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo...
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...
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...
1) A particle has a rest mass of 6.95×10−27 kg and a momentum of 5.15×10−18 kg⋅m/s....
1) A particle has a rest mass of 6.95×10−27 kg and a momentum of 5.15×10−18 kg⋅m/s. Determine the total relativistic energy of the particle.E= JJ Find the ratio of the particle's relativistic kinetic energy to its rest energy. ??rest= 2) Estimate the difference Δtdiff in a 15000-s time interval as measured by a proper observer and a relative observer traveling on a commercial jetliner. Δ?diff= s 3) Suppose that you have found a way to convert the rest energy of...
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...
5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10)...
5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10) where 8, m and k are fixed real constants greater than zero. (total 10 points) (a) Write down the Euler-Lagrange equation of motion for this system, and interpret the resulting equation in terms of a known physical system. (1 point) (b) Find Hamiltonian via Legendre transformation. (1 point) (c) Show that q(t) and the corresponding canonical momentum p(t) can be found as follows for...
5. A particle of mass m slides without friction and starting from rest down a block...
5. A particle of mass m slides without friction and starting from rest down a block of mass M in which the surface is shaped into a quadrant of a circle of radius R. The block also slides without friction on a table top. Choose two suitable generalized coordinates a) Using the Lagrange-multiplier formalism, find the equations of motion of the mass and the block and the reaction force of the block on the mass (ie. the constraint force b)...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential,...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential, 'U(r)', where 'r' is the distance from the center of the potential. A positional vector and momentum vector of a particle are vec r' and "vec p', respectively. (\vec means vector symbol.) Q1) An angular momentum vector vec J' is defined as vec J = \vec r x \vec p. Show that \vec J is conserved in such a gravitational potential U(r) which depends...
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,...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
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)...
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...
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...
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...
5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10) where 8, m and k are fixed real constants greater than zero. (total 10 points) (a) Write down the Euler-Lagrange equation of motion for this system, and interpret the resulting equation in terms of a known physical system. (1 point) (b) Find Hamiltonian via Legendre transformation. (1 point) (c) Show that q(t) and the corresponding canonical momentum p(t) can be found as follows for...
5. A particle of mass m slides without friction and starting from rest down a block of mass M in which the surface is shaped into a quadrant of a circle of radius R. The block also slides without friction on a table top. Choose two suitable generalized coordinates a) Using the Lagrange-multiplier formalism, find the equations of motion of the mass and the block and the reaction force of the block on the mass (ie. the constraint force b)...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential, 'U(r)', where 'r' is the distance from the center of the potential. A positional vector and momentum vector of a particle are vec r' and "vec p', respectively. (\vec means vector symbol.) Q1) An angular momentum vector vec J' is defined as vec J = \vec r x \vec p. Show that \vec J is conserved in such a gravitational potential U(r) which depends...
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,...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
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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