Chapter 10, Problem 35P

At time $t=0$, a cart of mass n moving at an initial velocity $v_0$ impacts a spring of stiffness k, as illustrated in Fig. P10.35.

The resulting deformation $x\left(t\right)$ of the spring satisfies the following second-order differential equation and initial conditions:

$m\ddot{x}\left(t\right)+kx\left(t\right)=0,x\left(0\right)=0,\stackrel{\dot{}}{x}\left(0\right)=v_0.$ (a) Determine the total solution for $x\left(t\right)$, subject to the initial conditions.

(b) Plot one-hall cycle of the deformation $x\left(t\right)$, and clearly label both its maximum value and the time it takes to get there.

(c) Mark each of the following statements as true (T) or false (F):

_____________Increasing the spring stiffness k will increase the frequency of $x\left(t\right)$.

_____________Increasing the initial velocity $v_0$ will increase the frequency of $x\left(t\right)$.

_____________Increasing the mass in will decrease the frequency of $x\left(t\right)$.

_____________Doubling the velocity $v_0$ will double the amplitude of $x\left(t\right)$.

_____________Doubling the mass m will double the amplitude of $x\left(t\right)$ .