# How do you find the position function #x(t)# if you suppose that the mass in a mass-spring-dashpot system with #m=25#, #c=10# and #k=226# is set in motion with #x(0)=20# and #x'(0)=41#?

In those considerations that follow, the positive axis is oriented from down-up for forces, velocities and displacements.

The spring-dashpot reacts according to

Joining both equations we have

This is a linear non-homogeneous differential equation whose solution is given by

and

The homogeneous equation has the general solution

Substituting this generic solution we get at

so

(This can be demostrated but requires a lot of algebra.)

The general solution is given by

#{(x(0) = C_1-(m g)/k = 20), (dot x(0) = -C_1/5+3C_2=41) :}#

From this system, we obtain the constants

So finally

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To find the position function ( x(t) ) for the mass-spring-dashpot system, we first need to solve the second-order linear ordinary differential equation (ODE) governing the system's motion. The equation is given by:

[ m\frac{{d^2x}}{{dt^2}} + c\frac{{dx}}{{dt}} + kx = 0 ]

Given the values ( m = 25 ), ( c = 10 ), and ( k = 226 ), and the initial conditions ( x(0) = 20 ) and ( x'(0) = 41 ), we substitute these values into the differential equation.

The solution to this second-order ODE is a linear combination of exponential functions. The general form of the solution is:

[ x(t) = e^{rt} ]

where ( r ) is a constant.

To find ( r ), we substitute ( x(t) = e^{rt} ) into the differential equation:

[ m\frac{{d^2}}{{dt^2}}e^{rt} + c\frac{{d}}{{dt}}e^{rt} + ke^{rt} = 0 ]

After differentiation, we get:

[ m(r^2e^{rt}) + c(re^{rt}) + ke^{rt} = 0 ]

This simplifies to:

[ mr^2 + cr + k = 0 ]

Now, substitute the given values of ( m ), ( c ), and ( k ) into the equation:

[ 25r^2 + 10r + 226 = 0 ]

Solve this quadratic equation for ( r ). Once you have the values of ( r ), the solution for ( x(t) ) will be:

[ x(t) = Ae^{r_1t} + Be^{r_2t} ]

where ( A ) and ( B ) are constants determined by the initial conditions, and ( r_1 ) and ( r_2 ) are the roots of the quadratic equation obtained earlier.

Finally, use the initial conditions ( x(0) = 20 ) and ( x'(0) = 41 ) to solve for ( A ) and ( B ), and plug them back into the solution ( x(t) ).

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To find the position function ( x(t) ) for the mass-spring-dashpot system, you need to solve the second-order linear differential equation that governs the system. The equation is:

[ m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0 ]

Given ( m = 25 ), ( c = 10 ), and ( k = 226 ), and the initial conditions ( x(0) = 20 ) and ( x'(0) = 41 ), you can use these values to solve for ( x(t) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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