A particle of mass m is attached to a spring with a spring constant k. The...

A particle of mass m is attached to a spring with a spring constant k. The other end of the spring is forced to move in a circle in the x ? y plane of radius R and angular frequency ?. The particle itself can move in all 3 directions. Write down the Lagrangian, and derive the equations of motion.

Expert Solution

genius_generous answered 2 years ago

A mass m = 3.27 kg is attached to a spring of force constant k =...

A mass m = 3.27 kg is attached to a spring of force constant k = 60.9 N/m and set into oscillation on a horizontal frictionless surface by stretching it an amount A = 0.17 m from its equilibrium position and then releasing it. The figure below shows the oscillating mass and the particle on the associated reference circle at some time after its release. The reference circle has a radius A, and the particle traveling on the reference circle...

A mass m = 1 kg is attached to a spring with constant k = 4...

A mass m = 1 kg is attached to a spring with constant k = 4 N/m and a dashpot with variable damping coefficient c. If the mass is to be pulled 5 m beyond its equilibrium (stretching the spring) and released with zero velocity, what value of c ensures that the mass will pass through the equilibrium position and compress the spring exactly 1 m before reversing direction? c =

10. It has a mass m attached to a spring of elastic constant k and that...

10. It has a mass m attached to a spring of elastic constant k and that suffers a damping b proportional to the speed, being b = k / 2. If the initial amplitude is A: a) How long does it take until the amplitude is worth A / 2? b) How long does it take until half of the initial energy has dissipated?

A particle of mass m = 1 oscillates without friction attached to a spring with k...

A particle of mass m = 1 oscillates without friction attached to a spring with k = 4. The motion of the particle is driven by the external force F(t) = 3 t cos(t). Find the equation of motion and solve it. Discuss the physical meaning of the solution.

1. A block of mass m attached to a spring with spring constant k oscillates horizontally...

1. A block of mass m attached to a spring with spring constant k oscillates horizontally on a frictionless table. Its velocity is 20 cm/s when x = -5 cm. Taking m = 100 gm, and spring constant = 2.5 N/m, a) Find out the equations of position, velocity, and acceleration of the ball. Find also the total energy of the block when its velocity was 20 cm/s. b) Oscillating particles generate waves. What will be the equation of a...

A mass m=4 is attached to both a spring with spring constant k=145 and a dash-pot...

A mass m=4 is attached to both a spring with spring constant k=145 and a dash-pot with damping constant c=4. The ball is started in motion with initial position x0=1 and initial velocity v0=6 . Determine the position function x(t) . x(t)=? Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form x(t)=c1e^(-ρt)cos(ω1t-α1) . Determine c1, ω1, ρ and α1. c1=? ω1=? ρ=? α1=? Graph the function x(t) together...

A particle of mass 2.00 kg is attached to a spring with a force constant of...

A particle of mass 2.00 kg is attached to a spring with a force constant of 300 N/m. It is oscillating on a horizontal frictionless surface with an amplitude of 4.00 m. A 7.00 kg object is dropped vertically on top of the 2.00 kg object as it passes through its equilibrium point. The two objects stick together. (a) Does the amplitude of the vibrating system increase or decrease as a result of the collision? decreases increases no change (b)By...

A particle of mass 4.00 kg is attached to a spring with a force constant of...

A particle of mass 4.00 kg is attached to a spring with a force constant of 300 N/m. It is oscillating on a horizontal frictionless surface with an amplitude of 5.00 m. A 9.00 kg object is dropped vertically on top of the 4.00 kg object as it passes through its equilibrium point. The two objects stick together. a) By how much does the amplitude of the vibrating system change as a result of collision? b) By how much does...

a block of mass m=0.10 kg attached to a spring whose spring constant is k=2.5 N/m...

a block of mass m=0.10 kg attached to a spring whose spring constant is k=2.5 N/m . At t=0.2s, the displacement x=-0.3m, and the velocity v=-2.0m/s a) find the equation of displacement as a function of time b) sketch the displacement as a function of time for the first cycle starting t=0s

A block with a mass M is attached to a horizontal spring with a spring constant...

A block with a mass M is attached to a horizontal spring with a spring constant k. Then attached to this block is a pendulum with a very light string holding a mass m attached to it. What are the two equations of motion? (b) What would these equations be if we assumed small x and φ? (Do note that these equations will turn out a little messy, and in fact, the two equations involve both variables (i.e. they are...

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