7) A spring has is stretched horizontally to the right from a wall 0.5 meters by...

7) A spring has is stretched horizontally to the right from a wall 0.5 meters by a force of 40 newtons. This spring has a mass of 10 kg. The mass is pulled to the right (from the equilibrium position) 1 meter and released (at t = 0) with an initial velocity to the right of 2 meters per second.

I) Write down the differential equation that governs the motion of the mass. Include the initial conditions. (assume no damping).

II) Find the position of the mass as a function of t for all t ≥ 0.

III) Write the solution in the form x(t) = A cos(ωt - δ)

IV) Suppose a damping device is attached to the mass-spring system with the magnitude of the damping force equal to c timesthe speed of the mass. For which values of c is the system over damped?

Expert Solution

Harmonic motion of a spring mass system. For an assumed damping force, proportional to velocity calculation of the constant of proportionality for which the system is overdamped.

genius_generous answered 2 years ago

Consider a mass-spring-dashpot system with mass 5kg, a spring which is stretched 3 meters by a...

Consider a mass-spring-dashpot system with mass 5kg, a spring which is stretched 3 meters by a force of 10N, and a dashpot which provides a 4N resistance for each m/s of velocity. The mass is also acted on by a periodic force 5 cos(ωt) for some number ω. • For what value of ω (if any) does practical resonance occur? • If the mass starts at rest, at equilibrium, find a formula for x(t) (in terms of ω), the distance...

Consider a mass-spring-dashpot system with mass 5kg, a spring which is stretched 6 meters by a...

Consider a mass-spring-dashpot system with mass 5kg, a spring which is stretched 6 meters by a force of 20N, and a dashpot which provides 4N resistance for each m/s of velocity. The mass is also acted on by a periodic force 5 cos(ωt) for some number ω. For what value of ω (if any) does practical resonance occur? If the mass starts at rest, at equilibrium, find a formula for x(t) in terms of ω, the distance between the mass...

A spring with a 6-kg mass and a damping constant 5 can be stretched 1.5 meters...

A spring with a 6-kg mass and a damping constant 5 can be stretched 1.5 meters beyond its natural length and held at rest there by a force of 7.5 newtons. Suppose the spring is stretched 3 meters below spring-mass equilibrium and then released with zero velocity. Find the position of the mass after t seconds. y(t)=? Please show all the work.

A 10kg weight is attached to a wall horizontally via a spring. The rest (equilibrium) length...

A 10kg weight is attached to a wall horizontally via a spring. The rest (equilibrium) length of the spring is 0.2m. The spring has stiffness coefficient k = 50. The weight drags along the ground, giving an effective damping coefficient of b = 40. We will ignore gravity and all other outside forces on the spring. If the weight is pulled to a position 0.5m away from the wall (thus +0.3m past equilibrium) and then flicked toward the wall at...

Differential Equations. A 10kg weight is attached to a wall horizontally via a spring. The rest...

Differential Equations. A 10kg weight is attached to a wall horizontally via a spring. The rest (equilibrium) length of the spring is 0.2m. The spring has stiffness coefficient k = 50. The weight drags along the ground, giving an effective damping coefficient of b = 40. We will ignore gravity and all other outside forces on the spring. If the weight is pulled to a position 0.5m away from the wall (thus +0.3m past equilibrium) and then flicked toward the...

A mass m, stretches horizontally from a spring which has spring constant k, and is subjected...

A mass m, stretches horizontally from a spring which has spring constant k, and is subjected to a retarding force equal to bv, where v is the instantaneous velocity of the mass. A particle of mass 10 gm moves along the x axis under the influence of two forces. The first is a force (in g cm s-2) of attraction to the origin O which is 40 times the distance from O. The second is a damping force proportional to...

1)A force of 2 N stretches a spring 0.5 meters. The mass of 1 kg is...

1)A force of 2 N stretches a spring 0.5 meters. The mass of 1 kg is attached to the spring and set into motion in a medium that offers a damped force equal 4 times the velocity. If the mass is at stating from 0.5 m above the equilibrium position with a downward initial velocity of sec/2.0m a) Find the equation for the position if the system is exerted by an external force of f(t)=4cos(t) b)Estimate the position of the...

a. A force of 40 lb is necessary to hold a spring that has been stretched...

a. A force of 40 lb is necessary to hold a spring that has been stretched from its natural length of 10 ft to a length of 15 ft. How much work is done in stretching the spring from 15 ft to a length of 18 ft? (Hint: Use Hook’s Law F = kx, where x represents distance that the spring is stretched or compressed past its natural length) b. A tank full of water has the shape of an...

A mass m = 1 kg is suspended from a spring that is stretched 1 cm...

A mass m = 1 kg is suspended from a spring that is stretched 1 cm under the influence of the weight of this mass. Now a periodic force is applied external of F (t) = 200 cos (vt) on the mass, which was initially in static balance. Disregarding all friction, get a relationship for position of the mass as a function of time, x (t). Also determine the value of ω which will cause resonance to occur

A game on The Price is Right has the contestant choose 12 tiles from a wall...

A game on The Price is Right has the contestant choose 12 tiles from a wall containing 25 tiles. Of the 25 tiles, 12 have a dollar sign on the back and the rest are blank. If all 12 of the tiles the constestant picks have a dollar sign, the contestant wins $100,000. However, if they pick just 10 tiles with the dollar sign, they win $10,000. In how many ways could the contestant pick exactly 10 of the 12...

Question