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5. You are the mechanical engineer in charge of machinery in a construction site. In a crane lowering a heavy load having a mass of (11.282) kg shown in the Figure 5.1, the cable connecting the crane and the load was suddenly jammed when the load was being lowered. Subsequently, the elasticity in the cable, having a spring constant of 1.06x10° N/m, caused the load to move in a vertical oscillatory (up-and-down) motion. If the load and cable in this situation can be thought of as a vertical mass-spring system as in Figure 5.2, the following equation can be derived: ma(t) + kx(t) = 0 (1) where wwwwwww m = mass of the load a(t) = acceleration of the load at time t k = spring constant of the cable x(t) = displacement of the load at time t. Get the value of a = 22 a. Knowing that the acceleration is the second derivative of displacement, formulate a second-order differential equation using equation 1. b. Solve the equation you formulated in (a) and find the general solution for the displacement, x(t). c. At the time of the jam (t = 0), if the load was being lowered at a speed of 0.1 m/s and if it position is considered to be at x(t) = 0, find the particular solution. d. Verify the solution you obtained in (c) by solving the equation you formulated in (a) using Laplace transforms. e. Given that the maximum possible displacement of the load before cable breakage is 0.12 m, determine whether the cable will break in this situation.

5. You are the mechanical engineer in charge of machinery in a construction site. In a crane lowering a heavy load having a mass of (11.282) kg shown in the Figure 5.1, the cable connecting the crane and the load was suddenly jammed when the load was being lowered. Subsequently, the elasticity in the cable, having a spring constant of 1.06x10° N/m, caused the load to move in a vertical oscillatory (up-and-down) motion. If the load and cable in this situation can be thought of as a vertical mass-spring system as in Figure 5.2, the following equation can be derived: ma(t) + kx(t) = 0 (1) where wwwwwww m = mass of the load a(t) = acceleration of the load at time t k = spring constant of the cable x(t) = displacement of the load at time t. Get the value of a = 22 a. Knowing that the acceleration is the second derivative of displacement, formulate a second-order differential equation using equation 1. b. Solve the equation you formulated in (a) and find the general solution for the displacement, x(t). c. At the time of the jam (t = 0), if the load was being lowered at a speed of 0.1 m/s and if it position is considered to be at x(t) = 0, find the particular solution. d. Verify the solution you obtained in (c) by solving the equation you formulated in (a) using Laplace transforms. e. Given that the maximum possible displacement of the load before cable breakage is 0.12 m, determine whether the cable will break in this situation.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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only need part d and part e

5. You are the mechanical engineer in charge of machinery in a construction site. In a crane lowering a heavy load having a mass of (11.282) kg shown in the Figure 5.1, the cable connecting the crane and the load was suddenly jammed when the load was being lowered. Subsequently, the elasticity in the cable, having a spring constant of 1.06x10° N/m, caused the load to move in a vertical oscillatory (up-and-down) motion. If the load and cable in this situation can be thought of as a vertical mass-spring system as in Figure 5.2, the following equation can be derived: ma(t) + kx(t) = o (1) where m = mass of the load a(t) = acceleration of the load at time t k = spring constant of the cable x(t) = displacement of the load at time t. Get the value of a = 22 a. Knowing that the acceleration is the second derivative of displacement, formulate a second-order differential equation using equation 1. b. Solve the equation you formulated in (a) and find the general solution for the displacement, x(t). c. At the time of the jam (t = 0), if the load was being lowered at a speed of 0.1 m/s and if its position is considered to be at x(t) = 0, find the particular solution. d. Verify the solution you obtained in (c) by solving the equation you formulated in (a) using Laplace transforms. е. Given that the maximum possible displacement of the load before cable breakage is 0.12 m, determine whether the cable will break in this situation.
Transcribed Image Text:5. You are the mechanical engineer in charge of machinery in a construction site. In a crane lowering a heavy load having a mass of (11.282) kg shown in the Figure 5.1, the cable connecting the crane and the load was suddenly jammed when the load was being lowered. Subsequently, the elasticity in the cable, having a spring constant of 1.06x10° N/m, caused the load to move in a vertical oscillatory (up-and-down) motion. If the load and cable in this situation can be thought of as a vertical mass-spring system as in Figure 5.2, the following equation can be derived: ma(t) + kx(t) = o (1) where m = mass of the load a(t) = acceleration of the load at time t k = spring constant of the cable x(t) = displacement of the load at time t. Get the value of a = 22 a. Knowing that the acceleration is the second derivative of displacement, formulate a second-order differential equation using equation 1. b. Solve the equation you formulated in (a) and find the general solution for the displacement, x(t). c. At the time of the jam (t = 0), if the load was being lowered at a speed of 0.1 m/s and if its position is considered to be at x(t) = 0, find the particular solution. d. Verify the solution you obtained in (c) by solving the equation you formulated in (a) using Laplace transforms. е. Given that the maximum possible displacement of the load before cable breakage is 0.12 m, determine whether the cable will break in this situation.
k = 1.06 x 106 N/m |x(t) x(t) m = (11.26 + 0.001a) kg Figure 5.1 Figure 5.2 f. During the oscillation, a structural failure in the crane causes an additional constant force f of 3.11x104 N to be exerted on the load. The forced oscillation can be expressed by the following equation: ma(t) + kx(t) = f (2) Using equation 2, formulate a second-order differential equation to represent the forced oscillation. P.T.O. g. Solve the equation you formulated in (f) and find the particular solution for the displacement, x(t).
Transcribed Image Text:k = 1.06 x 106 N/m |x(t) x(t) m = (11.26 + 0.001a) kg Figure 5.1 Figure 5.2 f. During the oscillation, a structural failure in the crane causes an additional constant force f of 3.11x104 N to be exerted on the load. The forced oscillation can be expressed by the following equation: ma(t) + kx(t) = f (2) Using equation 2, formulate a second-order differential equation to represent the forced oscillation. P.T.O. g. Solve the equation you formulated in (f) and find the particular solution for the displacement, x(t).
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