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Some particle of mass m is constrained to move on the surface of a vertical cylinder defined by 2 2 2 x ? y ? R , where R is the radius of the cylinder. The particle is subject to a force directed towards the origin and proportional to the distance of the particle from the origin F ?kr. The particle is also subject to gravitational force.
a) Find Lagrangian for this particle.b) Obtain Lagrange’s equations of motion for this particle.c) Find Hamiltonian for this particle.d) Obtain Hamilton’s canonical equations of motion for this particlee) Show that the angular momentum of this particle about the z-axis is conserved.f) Use Lagrange’s multiplier method to find the force of the constrain acting on this particle.g) Find the equilibrium position for this particle.h) Find the period of small oscillations of this particle near the equilibrium position.
The distance between two telephone poles is 50m. When a 1kg bird lands on the telephone wire midway between the poles, the wire sags .200m. How much tension does the bird produce in the wire? Ignore the weight of the wire.
A lab technician uses laser light with a wavelength of 670 nm to test a diffraction grating. When the grating is 40.2 cm from the screen, the first-order maxima appear 6.02 cm from the center of the pattern. How many lines per millimeter does this gr..
A 5 g bullet with speed of 400m/s gets stuck in a 1 kg wooden block (originally not moving). Classify the collision. (b) Find the new speed of block+bullet. (c) Find the amount of kinetic energy lost. Where does it go?
A uniform disk of radius 0.471 m and unknown mass is constrained to rotate about a perpendicular axis through its center. Find the mass of the disk
A 2.0 kg canister is moving in the positive x direction with an initial velocity of 4.0 m/s, How fast will the canstier be moving as a result of this force
A small, spherical asteroid has radius 4.00km and density 4760kg/m3. What's the escape speed from the surface
The basic function of an automobile's carburetor is to atomize the gasoline and mix it with air to promote rapid combustion
The Kinetic coeffficient of friction for a 12Kg mass on a table is 0.25. A string from the mass goes across a pulley and is attached to an 8.0kg mass hanging over the side of the table. find the magnitude of the acceleration of each mass
The coefficient of static friction between the ladder and the ground is 0.55, and the ladder makes a 50.0° angle with the ground. Explain how far up the ladder can an 760 N person climb before the ladder begins to slip?
A balloon is filled with helium at a pressure of 2.4 x 10^5 pa. The balloon is at a temperature of 18 degrees C and has a radius of 0.25 m. By what factor does the radius of the balloon increase
The weight of passengers on a roller coaster increases by 57 as the car goes through a dip with a 39 radius of curvature. What is the car's speed at the bottom of the dip
A 1500-kg car starts from rest to reach 60.0 miles/h in 7.50 s. What is the engine's power output and How much work is done by the engine during the acceleration
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