4. A Hooke's law force \( -k x \) and a constant conservative force \( F \) in the \( +x \)-direction act on an atomic ion with mass \( m \). (1) Show that a possible potential-energy function for this combination of forces is (2) \( x^{2} / 2-F x-F^{2} / 2 k \). (3) (a) If the total energy is \( E=F^{2} / k \), what are the maximum and minimum values of \( x \) that the ion reaches in its motion? (b) If the total energy is \( E=F^{2} / k \), what is its maximum speed? (c) For what value of \( x \) is the speed maximum?
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A Hooke's law force $-k x$ and a constant conservative force $F$ in the $+x$ -direction act on an atomic ion. (a) Show that a possible potential-energy function for this combination of forces is $U(x)=\frac{1}{2} k x^{2}-F x-F^{2} / 2 k$ . Is this the only possible function? Explain. (b) Find the stable equilibrium position. (c) Graph $U(x)$ (in units of $F^{2} / k )$ versus $x$ (in units of $F / k )$ for values of $x$ between $-5 F / k$ and 5$F / k$ . (d) Are there any unstable equilibrium positions? (e) If the total energy is $E=F^{2} | k,$ what are the maximum and minimum valnes of $x$ that the ion reaches in its motion? If the ion has mass $m,$ find its maximum speed if the total energy is $E=F^{2} / k .$ For what value of $x$ is the speed maximum?
The potential energy of a $1 \mathrm{~kg}$ particle free to move along the $x$ -axis is given by: $$ V(x)=\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right) \mathrm{J} $$ The total mechanical energy of the particle is $2 \mathrm{~J}$ Then, the maximum speed (in $\mathrm{m} / \mathrm{s}$ ) is: (a) 2 (b) $\frac{3}{\sqrt{2}}$ (d) $\frac{1}{\sqrt{2}}$ (c) $\sqrt{2}$
WORK, ENERGY AND POWER
Work, Energy, and Power
The potential energy of a $1 \mathrm{~kg}$ particle free to move along the $x$-axis is given by $V(x)=\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right) \mathrm{J}$. The total mechanical energy of the particle is $2 \mathrm{~J}$. Then, the maximum speed (in $\mathrm{m} / \mathrm{s}$ ) is [2006] (A) $\frac{3}{\sqrt{2}}$ (B) $\sqrt{2}$ (C) $\frac{1}{\sqrt{2}}$ (D) 2
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