1. A 1 kilogram mass is attached to a spring with a spring constant of 4...

1. A 1 kilogram mass is attached to a spring with a spring constant of 4 N/m. Write the equation of motion if the spring is stretched 25 cm below the equilibrium position and released.

a) Suppose the system experiences a constant forcing function downwards of 4 N. (Note that one would need to divide by the mass, but the mass is 1 kg so we don’t see a difference here.) Solve the non-homogeneous equation. (Keep everything else about the initial value problem the same.) How does the forcing function change the solution/motion?

b) Suppose the forcing function is the periodic function ?(?) = 4 cos ?. Solve that nonhomogeneous equation. What is the period and amplitude of the motion in this system?

c) Suppose the forcing function is the periodic function ?(?) = 4 cos 2?. Solve this nonhomogeneous equation. How does the period and amplitude of the motion change? How would you describe the motion?

Expert Solution

Kindly go through the solution and let me know in case of any doubt or further clarification in the comment box.
Thanks for the question :)
And your upvote will be really appreciable ;)

genius_generous answered 1 year ago

A 1-kilogram mass is attached to a spring whose constant is 16 N/m, and the entire...

A 1-kilogram mass is attached to a spring whose constant is 16 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneous velocity. The mass is initially released from rest from a point 1 meter below the equilibrium position. 1. Show the Diagram, given and required only ( No need to solve, Thumbs up will only be given if 1 is followed.)

A mass of 2 kilogram is attached to a spring whose constant is 8 N/m, and...

A mass of 2 kilogram is attached to a spring whose constant is 8 N/m, and the entire system is then submerged in a liquid that imparts a damping force equal to 8 times the instantaneous velocity. At t = 0 the mass is released from the equilibrium position with no initial velocity. An external force f(t) = 2U(t−1)*e^(−2(t−1)) is applied. (a) Write f(t), the external force, as a piecewise function and sketch its graph. (b) Write the initial-value problem....

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 =

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.

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...

A block with a mass of 0.488 kg is attached to a spring of spring constant...

A block with a mass of 0.488 kg is attached to a spring of spring constant 428 N/m. It is sitting at equilibrium. You then pull the block down 5.10 cm from equilibrium and let go. What is the amplitude of the oscillation? A block with a mass of 0.976 kg is attached to a spring of spring constant 428 N/m. It is sitting at equilibrium. You then pull the block down 5.10 cm from equilibrium and let go. What...

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...

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...

1. A mass of 0.019 kg attached to a spring with spring constant 27.0 N/m is...

1. A mass of 0.019 kg attached to a spring with spring constant 27.0 N/m is pulled to the right 8.0 cm and released. The mass oscillates with a frequency of 6.0 Hz. If the mass is pulled to the right 16.0 cm before being released, what is the frequency? a. 6.0 Hz b. 3.0 Hz c. 1.5 Hz d. 12 Hz e. 24 Hz 2. A window loses power/heat energy through a pane of glass to the cold outside....

A block of mass m = 2.5 kg is attached to a spring with spring constant...

A block of mass m = 2.5 kg is attached to a spring with spring constant k = 640 N/m. It is initially at rest on an inclined plane that is at an angle of θ = 27° with respect to the horizontal, and the coefficient of kinetic friction between the block and the plane is μk = 0.11. In the initial position, where the spring is compressed by a distance of d = 0.19 m, the mass is at...

Question