A weight of 45 N stretches a spring 5 cm. At time t = 0, the weight… A weight of 45 N st

A weight of 45 N stretches a spring 5 cm. At time t = 0, the weight… A weight of 45 N stretches a spring 5 cm. At time t = 0, the weight is released from its equilibrium position with an upward velocity of 28 cm s?1 . Determine the displacement x(t) that describes the subsequent free motionImage transcription textits equilibrium position with an upwardvelocity of 28 cm s. Determine thedisplacement r(t) that describ… Show more… Show more2)Consider a solid cylinder of radius a that is partially submerged in a bath of pure water. Let us find the motion of this cylinder in the vertical direction assuming that it remains in an upright position. If the displacement of the cylinder from its static equilibrium position is x, the weight of water displaced equals where ?w is the density of the water and g is the gravitational acceleration. This is the restoring force according to the Archimedes principle. The mass of the cylinder is Ah?, where ? is the density of cylinder.  4. A 4-kg mass is suspended from a 100 N/m spring. The mass is set in motion by giving it an initial downward velocity of 5 m/s from its equilibrium position. Find the displacement as a function of time.  5. A spring hangs vertically. A weight of mass M kg stretches it L m. This weight is removed. A body weighing m kg is then attached and allowed to come to rest. It is then pulled down 0 m and released with a velocity v0. Find the displacement of the body from its point of rest and its velocity at any time t.  6. A particle of mass m moving in a straight line is repelled from the origin by a force F. (a) If the force is proportional to the distance from the origin, find the position of the particle as a function of time. (b) If the initial velocity of the particle is a ? k, where k is the proportionality constant and a is the distance from the origin, find the position of the particle as a function of time. What happens if m < 1 and m = 1?  7)Image transcription text(4.4.40] Consequently, A = Ccos(p)and B = Csin(). Once again, we obtainC = VA- + B'. On the other... Show more... Show more8)A body with mass m = 1 2 kg is attached to the end of a spring that is stretched 2 m by a force of 100 N. Furthermore, there is also attached a dashpot4 that provides 6 N of resistance for each m/s of velocity. If the mass is set in motion by further stretching the spring 12 m and giving it an upward velocity of 10 m/s, let us find the subsequent motion. We begin by first computing the constants. The spring constant is k = (100 N)/(2 m) = 50 N/m. Therefore, the differential equation is 1 2 x ?? + 6x ? + 50x = 0 9)In its simplest form a wind vane is a flat plate or airfoil that can rotate about a vertical shaft In static equilibrium it points into the wind. There is usually a counterweight to balance the vane about the vertical shaft. A vane uses a combination of the lift and drag forces on the vane to align itself with the wind. As the wind shifts direction from 0 to the new direction ?i , the direction ? in which the vane currently points is governed by the equation 10) For a fixed value of ?/?, what is the minimum number of cycles required to produce a reduction of at least 50% in the maxima of an underdamped oscillator?  11). For what values of c does x ?? + cx? + 4x = 0 have critically damped solutions? 6. For what values of c are the motions governed by 4x ?? + cx? + 9x = 0 (a) overdamped, (b) underdamped, and (c) critically damped?  12). For an overdamped mass-spring system, prove that the mass can pass through its equilibrium position x = 0 at most once.13)Image transcription textFor the following values of m. 8, and k,find the position (t) of a dampedoscillator for the given init... Show more... Show more  Math MATH 133 Share QuestionEmailCopy link