AP Physics C Mechanics 3.4: Conservation of Mechanical Energy

Use mechanical-energy conservation to connect kinetic and potential energy, while accounting for work by nonconservative interactions.

[Maximum number: 4]

In Scenario 1, a system composed of two springs, A and B, and a block of mass m is at rest on

a horizontal surface. Friction between the block and the surface is negligible. Each spring is

attached to a fixed wall and the block, as shown in Figure 1. Spring A has a spring constant k

and Spring B has a spring constant 2 k. Each spring is at its relaxed length when the block is at

position x=0, as shown.

Figure 1

The block is moved to $x=x_{1}$ and held at rest, as shown in Figure 2.

Figure 2

The block is released from rest at $x=x_{1}$ and begins to oscillate. Derive an expression for the

speed v of the block as the block passes through $x=\frac{1}{2} x_{1}$ . Express your answer in terms of m,

$k, x_{1}$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental

physics principle or an equation from the reference information.

\multirow[t]{4}{*}{B} & For a multistep derivation that includes energy conservation or simple harmonic motion & Point B1 \\ \hline & For relating the presence of both springs to the behavior of the system & Point B2 \\ \hline & For relating positions $x=x_{1}$ and $x=\frac{1}{2} x_{1}$ to the oscillation of the block & Point B3 \\ \hline & For a correct expression for v in terms of given quantities & Point B4 \\ \hline \end{tabular}

Example Responses

		$x=x_{\text {max }} \cos (\omega t+\phi)$
$E_{\text {tot }, 0}=U_{\mathrm{A}}\left(x_{1}\right)+U_{\mathrm{B}}\left(x_{1}\right)+K_{\text {block }}\left(x_{1}\right)$
$E_{\text {tot }, f}=U_{\mathrm{A}}\left(\frac{1}{2} x_{1}\right)+U_{\mathrm{B}}\left(\frac{1}{2} x_{1}\right)+K_{\text {block }}\left(\frac{1}{2} x_{1}\right)$		$\begin{aligned} & x(t)=x_{1} \cos \omega t & v(t)=-x_{1} \omega \sin \omega t \end{aligned}$
$\begin{aligned} & E_{\mathrm{tot}, 0}=\frac{1}{2}(k)\left(x_{1}\right)^{2}+\frac{1}{2}(2 k)\left(x_{1}\right)^{2}+0 & E_{\mathrm{tot}, f}=\frac{1}{2}(k)\left(\frac{1}{2} x_{1}\right)^{2}+\frac{1}{2}(2 k)\left(\frac{1}{2} x_{1}\right)^{2}+K_{\mathrm{block}, \frac{1}{2} x_{1}} \end{aligned}$	OR	$\begin{aligned} & k_{\text {eff }}=k+2 k=3 k & \omega=\sqrt{\frac{3 k}{m}} \end{aligned}$
$\begin{aligned} & E_{\text {tot }, 0}=E_{\text {tot }, f} & \frac{1}{2} k x_{1}^{2}+k x_{1}^{2}=\frac{1}{8} k x_{1}^{2}+\frac{1}{4} k x_{1}^{2}+K_{\text {block }, \frac{1}{2} x_{1}} & \frac{3}{2} k x_{1}^{2}=\frac{3}{8} k x_{1}^{2}+K_{\text {block, } \frac{1}{2} x_{1}} \end{aligned}$		$\begin{aligned} & \frac{x_{1}}{2}=x_{1} \cos \omega t_{\frac{1}{2} x_{1}} & \cos \omega t_{\frac{1}{2} x_{1}}=\frac{1}{2} & t_{\frac{1}{2} x_{1}}=\frac{\pi}{3 \omega} \end{aligned}$
$\begin{aligned} & K_{\text {block }, \frac{1}{2} x_{1}}=\frac{9}{8} k x_{1}^{2} & \frac{1}{2} m v^{2}=\frac{9}{8} k x_{1}^{2} \end{aligned}$		$\begin{aligned} v_{\frac{1}{2} x_{1}} & =-x_{1} \omega \sin \omega t_{\frac{1}{2} x_{1}} v_{\frac{1}{2} x_{1}} & =-x_{1} \sqrt{\frac{3 k}{m}} \sin \frac{\pi}{3} \end{aligned}$
$v=\frac{3}{2} x_{1} \sqrt{\frac{k}{m}}$		$v=\left\|v_{\frac{1}{2}} x_{1}\right\|=\frac{3}{2} x_{1} \sqrt{\frac{k}{m}}$

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3.4.B—Describe the behavior of a system using conservation of mechanical energy principles question 1