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Problem 5 Consider the motion of a particle in a potentail V(r) = kr. Using the...
2. Consider the effective potential for an inverse-square force (the Kepler problem). Find the frequency of...
2. Consider the effective potential for an inverse-square force (the Kepler problem). Find the frequency of small radial oscillations about the minimum (which corresponds to circular motion). Show that this frequency is exactly equal to the frequency of revo- lution for a circular orbit
Problem 10.10 meProblem 10.10 Consider a central force given by F(r) = -K/r3 with K>O. Plot...
Problem 10.10
meProblem 10.10 Consider a central force given by F(r) = -K/r3 with K>O. Plot the effective potential and discuss possible types of motion.
meProblem 10.10 Consider a central force given by F(r) = -K/r3 with K>O. Plot the effective potential and discuss possible types of motion.
Earth ny In. 2. Using Lagrange's equation, write the equations of motion of the spacecraft (particle Q) in problem 4. The potential energy of a particle in Earth's central gravity field is: V...
Earth ny In. 2. Using Lagrange's equation, write the equations of motion of the spacecraft (particle Q) in problem 4. The potential energy of a particle in Earth's central gravity field is: V The negative sign arises because the gravity potential is defined as zero at r-o The resulting equations of motion should be the same as those in problem 4. G M m
Earth ny In.
2. Using Lagrange's equation, write the equations of motion of the spacecraft (particle...
# 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...
Understand how to find the equation of motion of a particle undergoing uniform circular motion. Consider...
Understand how to find the equation of motion of a particle undergoing uniform circular motion. Consider a particle--the small red block in the figure--that is constrained to move in a circle of radius R. We can specify its position solely by θ(t), the angle that the vector from the origin to the block makes with our chosen reference axis at time t. Following the standard conventions we measure θ(t) in the counterclockwise direction from the positive x axis. (Figure 1)...
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...
A particle of mass 5 kg is subject to a conservative force whose potential energy (in...
A particle of mass 5 kg is subject to a conservative force whose potential energy (in joules) as a function of position (in meters) is given by the equation U(x) =-100x5e-1x [where x > 0] (a) Determine the position xo where the particle experiences stable equilibrium (b) Find the potential energy Uo of the particle at the position x 2106 The particle is displaced slightly from position x = xo and released (c) Determine the effective value of the spring...
In Schwarzschild spacetime, consider a stationary observer at position r-10m, 0 (a) A freely movi...
In Schwarzschild spacetime, consider a stationary observer at position r-10m, 0 (a) A freely moving particle passing the observer has radial and angular velocity components 1/2 and1/4 (as measured by the observer). Find the equation of motion of the particle in terms of r(r) and φ(r). Do not solve the equation. Describe qualitatively the shape of the particle's trajectory by using the effective potential.
In Schwarzschild spacetime, consider a stationary observer at position r-10m, 0 (a) A freely moving particle...
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...
Problem 2 Consider a particle on a sphere of fixed radius r and V = 0...
Problem 2 Consider a particle on a sphere of fixed radius r and V = 0 on the spherical surface. The particle is found in either of the following wavefunctions: Y1,1(0,0) or Y1,-1(0,0). a) What is the energy difference between the two possible wavefunctions? b) Determine the magnitude of angular momentum and its projection onto the z-axis for both wavefunctions. c) Do the energy and the magnitude of angular momentum change when the radius r of the sphere is increased?...
2. Consider the effective potential for an inverse-square force (the Kepler problem). Find the frequency of small radial oscillations about the minimum (which corresponds to circular motion). Show that this frequency is exactly equal to the frequency of revo- lution for a circular orbit
Problem 10.10
meProblem 10.10 Consider a central force given by F(r) = -K/r3 with K>O. Plot the effective potential and discuss possible types of motion.
meProblem 10.10 Consider a central force given by F(r) = -K/r3 with K>O. Plot the effective potential and discuss possible types of motion.
Earth ny In. 2. Using Lagrange's equation, write the equations of motion of the spacecraft (particle Q) in problem 4. The potential energy of a particle in Earth's central gravity field is: V The negative sign arises because the gravity potential is defined as zero at r-o The resulting equations of motion should be the same as those in problem 4. G M m
Earth ny In.
2. Using Lagrange's equation, write the equations of motion of the spacecraft (particle...
# 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) 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...
A particle of mass 5 kg is subject to a conservative force whose potential energy (in joules) as a function of position (in meters) is given by the equation U(x) =-100x5e-1x [where x > 0] (a) Determine the position xo where the particle experiences stable equilibrium (b) Find the potential energy Uo of the particle at the position x 2106 The particle is displaced slightly from position x = xo and released (c) Determine the effective value of the spring...
In Schwarzschild spacetime, consider a stationary observer at position r-10m, 0 (a) A freely moving particle passing the observer has radial and angular velocity components 1/2 and1/4 (as measured by the observer). Find the equation of motion of the particle in terms of r(r) and φ(r). Do not solve the equation. Describe qualitatively the shape of the particle's trajectory by using the effective potential.
In Schwarzschild spacetime, consider a stationary observer at position r-10m, 0 (a) A freely moving particle...
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...
Problem 2 Consider a particle on a sphere of fixed radius r and V = 0 on the spherical surface. The particle is found in either of the following wavefunctions: Y1,1(0,0) or Y1,-1(0,0). a) What is the energy difference between the two possible wavefunctions? b) Determine the magnitude of angular momentum and its projection onto the z-axis for both wavefunctions. c) Do the energy and the magnitude of angular momentum change when the radius r of the sphere is increased?...
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