Question

Given the differential equation and initial conditions for a mass-spring system: (Do NOT solve this IVP.) x′" (t)+x′ (t)+1/4 x=0,x(0)=1/2 ,x"(0)=− 7/4

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Given that the solution of the problem is x(t)= 1/2 e⁻ᵗ/²⁻ᵗ/² − 3/2 te⁻ᵗ/²
a) Is the motion of the mass critically damped, over damped, or under damped? Why? (4 points) b) When, if ever, does the mass pass through the equilibrium position? Is the mass moving up or down at that time? ( 6 points) c) Find the time after t=0 at which the mass attains its extreme displacement from the equilibrium position. (5 points) d) Use the information above to sketch a reasonable graph of the motion 0 the mass.

Answer 1

The mass-spring system described by the differential equation and the given solution is critically damped. To determine when the mass passes the equilibrium position and finds the extreme displacement, one would set x(t) to 0 and find the maximum or minimum of the x(t) function, respectively.

Step-by-step explanation:

The student's question relates to a damped mass-spring system described by a differential equation and various conditions of motion. To determine the nature of damping, we look at the given solution of the differential equation:

x(t) = 1/2 e⁻¹/² - 3/2 te⁻¹/²

This solution indicates that the mass-spring system is critically damped because the expression does not contain any oscillatory (sinusoidal or cosinusoidal) terms, and it involves a term that's linear in time multiplied by an exponential decay, which is a hallmark of critical damping. To find when the mass passes through the equilibrium position, one has to solve for x(t) = 0 and determine whether the velocity is positive or negative at that time, indicating the direction of motion. The extreme displacement is found by locating the maximum or minimum point of the x(t) function, which can be determined by setting the first derivative of x(t) equal to zero and verifying that the second derivative is positive (for a minimum) or negative (for a maximum).

The graph of the motion would reflect the critical damping condition with the mass rapidly approaching the equilibrium position without oscillating.