Chapter 6

A girl of mass $49.0\mathrm{kg}$ is on a swing, which has a mass of $1.0\mathrm{kg}.$ Suppose you pull her back until her center of mass is $2.0\mathrm{m}$ above the ground. Then you let her $\mathrm{go},$ and she swings out and returns to the same point. Are all forces acting on the girl and swing conservative?

Can a potential energy function be defined for the force of friction?

Can the potential energy of a spring be negative?

One end of a rubber band is tied down and you pull on the other end to trace a complicated closed trajectory. If you were to measure the elastic force $F$ at every point and took its scalar product with the local displacements, $\overrightarrow{F}\cdot \mathrm{\Delta }\overrightarrow{r},$ and then summed all of these, what would you get?

Can a unique potential energy function be identified with a particular conservative force?

A projectile of mass $m$ is launched from the ground at $t=0$ with a speed $v_0$ and at an angle ${\theta }_0$ above the horizontal. Assuming that air resistance is negligible, write the kinetic, potential, and total energies of the projectile as explicit functions of time.

The energy height, $H$, of an aircraft of mass $m$ at altitude $h$ and with speed $v$ is defined as its total energy (with the zero of the potential energy taken at ground level) divided by its weight. Thus, the energy height is a quantity with units of length. a) Derive an expression for the energy height, $H$, in terms of the quantities $m,h$, and $v$. b) A Boeing 747 jet with mass $3.5\cdot {10}^5\mathrm{kg}$ is cruising in level flight at $250.0\mathrm{m}/\mathrm{s}$ at an altitude of $10.0\mathrm{km}.$ Calculate the value of its energy height. Note: The energy height is the maximum altitude an aircraft can reach by "zooming" (pulling into a vertical climb without changing the engine thrust). This maneuver is not recommended for a 747 , however.

A body of mass $m$ moves in one dimension under the influence of a force, $F(x)$, which depends only on the body's position. a) Prove that Newton's Second Law and the law of conservation of energy for this body are exactly equivalent. b) Explain, then, why the law of conservation of energy is considered to be of greater significance than Newton's Second Law.

The molecular bonding in a diatomic molecule such as the nitrogen $\left({\mathrm{N}}_2\right)$ molecule can be modeled by the Lennard Jones potential, which has the form $$U(x)=4U_0({\left(\frac{x_0}{x}\right)}^{12}-{\left(\frac{x_0}{x}\right)}^6)$$ where $x$ is the separation distance between the two nuclei and $x_0$, and $U_0$ are constants. Determine, in terms of these constants, the following: a) the corresponding force function; b) the equilibrium separation $x_0$, which is the value of $x$ for which the two atoms experience zero force from each other; and c) the nature of the interaction (repulsive or attractive) for separations larger and smaller than $x_0$.

What is the gravitational potential energy of a $2.0-\mathrm{kg}$ book $1.5\mathrm{m}$ above the floor?