Problem 5.87

พิจารณากราฟ f กับ x

f (N)

\(_{2\mu mg}\)

\(_{2\mu mg -}\frac{_{\mu mgx}}{_{\mu mg}^{\ell}}\)

x (m)

\(_{0x\ell}\)

จะได้ \(W_{f}=-\frac{1}{2}(2\mu mg+2\mu mg-\frac{\mu mgx}{\ell})x\)

\(W_{f}=\frac{\mu mgx^{2}}{2\ell}- 2\mu mgx\)

จาก \(\Sigma W_{NC}= \Delta E\)

\(W_{n}+ W_{f}= (K_{f}+ U_{f}) - (K_{i}+ U_{i})\)

\(0 +\frac{\mu mgx^{2}}{2\ell}-2\mu mgx=(0+0)-(\frac{1}{2}mu^{2}+0)\)

\(\mu gx^{2} - 4\mu g\ell x = -\ell u^{2}\)

\(\mu gx^{2} - 4\mu g\ell x + \ell u^{2} = 0\)

\(x =\frac{4\mu g\ell \pm 16\mu ^{2}g^{2}\ell ^{2}-4\mu g\ell u^{2}}{2\mu g}\)

\(x =\frac{4\mu g\ell  \pm  4\mu g\ell 1 -\frac{u^{2}}{4\mu g\ell}}{2\mu g}\)

\(x = 2\ell (1 \pm \sqrt{1 -\frac{u^{2}}{4\mu g\ell}})\)

\(\therefore x = 2\ell (1 -\sqrt{1 -\frac{u^{2}}{4\mu g\ell}})\)