A 10-kg mass is attached to a spring having a spring constant of 140 N/m. The mass is started in motion from the equilibrium position with an initial velocity of 1 m/sec in the upward direction and with an applied external force F(t) = 5 sin t. Find the subsequent motion of the mass if the force

Question

A 10-kg mass is attached to a spring having a spring constant of 140 N/m. The mass is started in motion from the equilibrium position with an initial velocity of 1 m/sec in the upward direction and with an applied external force F(t) = 5 sin t. Find the subsequent motion of the mass if the force due to air resistance is [latex]-90 \dot{x} N[/latex].

Accepted Answer

Here [latex]m=10, k=140, a=90[/latex], and F(t) = 5 sin t. The equation of motion, (14.1), becomes

[latex]\ddot{x}+9 \dot{x}+14 x=\frac{1}{2} \sin t[/latex] (1)

The general solution to the associated homogeneous equation [latex]\ddot{x}+9 \dot{x}+14 x=0[/latex] is (see Problem 14.5)

[latex]x_{h}=c_{1} e^{-2 t}+c_{2} e^{-7 t}[/latex]

Using the method of undetermined coefficients (see Chapter 11), we find

[latex]x_{p}=\frac{13}{500} \sin t-\frac{9}{500} \cos t[/latex] (2)

The general solution of (1) is therefore

[latex]x=x_{h}+x_{p}=c_{1} e^{-2 t}+c_{2} e^{-7 t}+\frac{13}{500} \sin t-\frac{9}{500} \cos t[/latex]

Applying the initial conditions, [latex]x(0)=0 ext { and } \dot{x}(0)=-1[/latex], we obtain

[latex]x=\frac{1}{500}\left(-90 e^{-2 t}+99 e^{-7 t}+13 \sin t-9 \cos t ight)[/latex]

Note that the exponential terms, which come from [latex]x_{h}[/latex] and hence represent an associated free overdamped motion, quickly die out. These terms are the transient part of the solution. The terms coming from [latex]x_{p}[/latex], however, do not die out as [latex]t ightarrow \infty[/latex]; they are the steady-state part of the solution.

__________________________________________________________________________________________________________________________

Eq. (14.1) : [latex]\ddot{x}+\frac{a}{m} \dot{x}+\frac{k}{m} x=\frac{F(t)}{m}[/latex]

Question 14.9: A 10-kg mass is attached to a spring having a spring constan...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions