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# Classical Mechanics

### Lagrangian and Hamiltonian's Mechanics: Write down the Lagrangian L for two particles of equal mass m1 = m2 = m, confined to the x-axis and connected by a spring with potential energy U = ½ k x^2. (Here x is the extension of the spring, x = (x1- x2 -l), where l is the spring's outstretched length, and that mass l remains to the right of mass 2 at all times.] (b) Rewrite L in terms of the new variables X = ½(x1+x2) (the CM position) and x (the extension) and write down the two Lagrange equation of X and x (c) Solve for X(t) and x(t) and describe the motion.

Lagrangian and Hamiltonian's Mechanics: Write down the Lagrangian L for two particles of equal mass m1 = m2 = m, confined to the x-axis and connected by a spring with potential energy U = ½ k x^2. (Here x is the extension of the spring, x = (x1- x2 -l), where l is the spring's outstretched length, and that mass l remains to

### Minimum Connecting Distance for Nodes

Given the following distances between destination nodes, what is the minimum distance that connects the nodes? Between nodes 1 - 2: 125; Between nodes 2 - 3: 150; Between nodes 1 - 3: 200 A. 125 B. 200 C. 275 D. 350 Given the following distances between destination nodes, what is the minimum distance that connects

### Meeting in a circle

Two people start at the same place and walk around a circular lake in opposite directions. One has an angular speed of 1.3 10-3 rad/s, while the other has an angular speed of 3.3 10-3 rad/s. How long will it be before they meet?

### Classical Mechanics: Lagrange of a simple pendulum.

A pendulum consists of a mass m suspended by a massless spring with unextended length of b and spring constant k. Find Lagrange's equation of motion. Assume that the pendulum is constrained to swing in a single plane.

### particle passes from potential U1 to potential U2

See attached file for full problem description. 7.8 Consider a region of space divided by a plane. The potential energy of a particle in region 01 is U1 and in region 02 it is U2. If a particle of mass m and with speed v1 in region 01 passes from region 01 to region 02 such that its path in region 01 makes an angle theta1

### amplitude and phase of Driven harmoic oscillators

See attached file for full problem description. 1. Driven harmoic oscillators Suppose that a driven harmoic oscillators with belta = 1/3w0 is driven with force F = F0cos(wt) with driven frequency w = 1/3w0. Find the amplitude 'D' and phase a of the motino x(t) = Dcos(wt- a). expressing them purely in terms of F0, k and nume

### Solid sphere and a cylinder kinetic energy

A solid sphere and a cylinder of the same mass and radius roll without slipping at the same speed. It is correct to say that the total kinetic energy of the solid sphere is more than the total kinetic energy of the cylinder. less than the total kinetic energy of the cylinder. equal to the total kinetic energy of

### Disk, Hoop, and a Sphere Rolling Down an Incline

A disk, a hoop, and a sphere are released at the same time at the top of an inclined plane. They all roll without slipping. In what order do they reach the bottom? A. disk, hoop, sphere B. sphere, disk, hoop C. hoop, sphere, disk D. hoop, disk, sphere.

### The A string on a string bass is tuned to vibrate at a fundamental frequency of 55.0 Hz. If the tension in the string were increased by a factor of four, what would be the new fundamental frequency?

The A string on a string bass is tuned to vibrate at a fundamental frequency of 55.0 Hz. If the tension in the string were increased by a factor of four, what would be the new fundamental frequency?

### Transverse wave meet on two parallel wires with tension

Please help with the following problem. Provide step by step calculations. Two wires are parallel, and one is directly above the other. Each has a length of 50.0 m and a mass per unit length of 0.020 kg/m. However, the tension in wire A is 6.00 x 10^2 N and the tension in wire B is 3.00 x 10^2 N. Transverse wave pulses ar

### Tension of a Rope

A tightrope walker with a mass of 60.0 kg stands at the center of a rope which was initially strung horizontally between two poles. His weight causes the rope to sag symmetrically, making an angle of 4.80 with the horizontal. What is the tension in the rope? 359 N 589 N 2470 N 3520 N A tightrope wal

### Oscillators and Classical Spring Concepts

See attachment for better formula representation. 1) Consider the two-dimensional anisotropic oscillator with motion given by: x(t) = Ax cos (ωxt) y(t) = Ay cos (ωyt - δ). (a) Prove that if the ratio of frequencies is rational (that is ωx / ωy = p/q where p and q are integers) then the motion is periodic. What i

### Various classical physics questions

See attached file for a full problem description with correct formatting. 1. Conversion factors: A Joule is the MKS unit for energy; the cgs (centimeter- gram-second) unit of energy is the erg. I have started the conversion from Joules to ergs for you. Write down conversion factors and convert the following MKS units to cg

### Sum of momenta of two joggers

A 85 kg jogger is heading due east at a speed of 2.0 m/s. A 55 kg jogger is heading 32 degrees north of east at a speed of 3.0 m/s. Find the magnitude and direction of the sum of the momenta of the two joggers.

### Tension in the String Used to Hang a Block Over a Pulley

A 9.00 kg hanging weight is connected to a string over a pulley to a 5.00 kg block that is sliding on a flat table (see figure P5.24, page 142). If the coefficient of kinetic friction is 0.15, find the tension in the string.

### Acceleration: Speed of the Car When It Went Over the Cliff

A car drives straight off the edge of a cliff that is 54 m high. The police at the scene of the accident note that the point of impact is 130 m from the base of the cliff. How fast was the car traveling when it went over the cliff?

### Modern Physics: What is the final speed of the probe?

A 5.0 x 104 kg space probe is traveling at a speed of 11000 m/s through deep space. Retrorockets are fired along the line of motion to reduce the probe/s speed. The retrorockets generate a force of 4.0 x 10^5 N over a distance of 2500 km. What is the final speed of the probe?

### What is the magnitude and direction of the tension that the right section of the wire sustains.?

A tree limb breaks off and comes to rest across a barbed wire fence at a point that is not in the middle between two fence posts. The limb exerts a downward force of 11 N on the wire. The left section of the wire makes an angle of 14 degree relative to the horizontal and sustains a tension of 447 N. What is the magnitude and dir

### Static Friction and Dual Block System

One block rests upon a horizontal surface. A second identical block rests upon the first one. The coefficient of static friction between the blocks is the same as the coefficient of static friction between the lower block and the horizontal surface. A horizontal force is applied to the upper block, and its magnitude is slowly in

### A truck covers 40.0 m in 8.50 s while smoothy slowing down to a final speed of 2.80m/s . Find orginal speed.

A truck covers 40.0 m in 8.50 s while smoothy slowing down to a final speed of 2.80m/s . Find orginal speed.

### Finding the Frequency of Oscillatory Motion

Two masses M=100gms. and Q = 200gms. slide freely on a horizontal, frictionless track and are connected by a spring whose force constant is k = .5 Newtons/meter. Find the frequency of oscillatory motion for this system.

### Determining Horizontal Distance

A horizontal rifle is fired at a bull's eye. The muzzle speed of the bullet is 670 m/s. The barrel is pointed directly at the center of the bull's eye, but the bullet strikes the target .025 m below the center. What is the horizontal distance between the end of the rifle and the bull's eye?

### Speed question

A diver runs horizontally with a speed of 1.20 m/s off a platform that is 10 m above the water. What is his speed just before striking the water?

### Questions involving different gravities and masses

15. The height achieved in a jump is determined by the initial vertical velocity that the jumper is able to achieve. Assuming that this is a ﬁxed number, how high can an athlete jump on Mars if she can clear 1.55 In on Earth? 22. The radius of a neutron star is T50 times smaller than Earth's radius, and its mass is 1.8 H 105

### Steel piano string: Fundamental frequency and wavelength, speed of a wave

A steel piano string is .05m in length, with a mass of .005kg, and has a tension of 400N. 1) what is the fundamental frequency and wavelength of the string? 2) What is the speed of a wave on this string? 3) If the tension is increased to 450N, what is the new fundamental frequency?

### Motion: Calculate the minimum average speed an opponent must move to hit a lofted ball

If you loft a ball with an initial speed of 15.0 m/s at an angle of 50 degrees above the horizontal. At this instant your opponent is 10 m away from the ball. He begins moving away from you .30 sec later, hoping to reach the ball and hit it back at the moment that it is 2.10 m above its launch point. I'm trying to figure out wha

### Pressure on the ground

A woman wearing high healed shoes placed about 50% of her full weight on a single heal when walking. A) Assuming the woman weighs 530 N, what is the pressure on the ground under one heal if the area of contact per heal is 6.5 cm^2? What if this area was 1.0 cm^2? B) How does this compare with the pressure underneath an ele

### Measurements of Beams

Please see attachment 1) Determine the location of the centroid yc, measured from the bottom, for a beam with the cross section shown in the figure. A) 0.5 in B) 2.5 in C) 3.0 in D) 5.0 in 2) Determine the maximum flexural stress produced by a resisting moment Mr of +5000 ft*lb if the beam has the cross section sho

### Extension Length of a Circus Performer's Elastic Rope

A 56.5 kg circus performer bobs up and down at the end of a long elastic rope at a rate of once every 5.70 s. The elastic rope obeys Hooke's law. Find how much the rope is extended beyond its unloaded length when the performer hangs at rest.

### Dynamics engineering problem

1. The balloon A is ascending at the rate of v_A=12km/h and is being carried horizontally by the wind at v_w=20km/h. If a ballast bag is dropped from the balloon at the instant h=50m, determine the time needed for it to strike the ground. Assume that the bag was released from the balloon with the same velocity as the balloon. Al