10. Conservation of Energy

In FIGURE EX10.27, what is the maximum speed of a 2.0 g particle that oscillates between x = 2.0 mm and x = 8.0 mm

A system in which only one particle moves has the potential energy shown in FIGURE EX10.31. What is the x-component of the force on the particle at x = 5, 15, and 25 cm?

A particle moving along the y-axis is in a system with potential energy U = 4y³ J, where y is in m. What is the y-component of the force on the particle at y = 0 m, 1 m, and 2 m?

A particle is constrained to move in one dimension along the x axis and is acted upon by a force given by $\overrightarrow{F}\left(x\right)$ = - (k/x³) î, where k is a constant with units appropriate to the SI system. Find the potential energy function U(x), if U is arbitrarily defined to be zero at x = 2.0m, so that U (2.0m) = 0.

CALC The potential energy for a particle that can move along the x-axis is U = Ax² + B sin(πx/L), where A, B, and L are constants. What is the force on the particle at (a) x = 0, (b) x = L/2, and (c) x = L?

In FIGURE EX10.26, What minimum speed does a 100 g particle need at point B to reach point A?

The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero. In which interval(s) of 𝓍 is the force on the particle to the right?

You learned in Chapter 41 that the binding energy of the electron in a hydrogen atom is 13.6 eV. By how much does the mass decrease when a hydrogen atom is formed from a proton and an electron? Give your answer both in atomic mass units and as a percentage of the mass of the hydrogen atom.

The potential energy of the two atoms in a diatomic (two-atom) molecule can be approximated as (Lennard-Jones potential) U(r) = -(a/r⁶) + (b/r¹²), where r is the distance between the two atoms and a and b are positive constants. At what values of r is U(r) a minimum? A maximum?

A particle moving along the x-axis is in a system that has potential energy U = x³ - 3x J, where x is in m. For each, is it a point of stable or unstable equilibrium?

A system has potential energy $U\left(x\right)=\left(10\text{J}\right)[1-\sin (\left(3.14\text{rad/m}\right)x\left)\right]$ as a particle moves over the range $0\text{ m }\le x\le 3\text{ m}$. For each, is it a point of stable or unstable equilibrium?

(III) The potential energy of the two atoms in a diatomic (two-atom) molecule can be approximated as (Lennard-Jones potential) U(r) = -(a/r⁶) + (b/r¹²), where r is the distance between the two atoms and a and b are positive constants. Let F be the force one atom exerts on the other. For what values of r is F > 0, F < 0, F = 0?

The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero. At what value(s) of 𝓍 is the magnitude of the force a minimum?

FIGURE EX10.24 is the potential-energy diagram for a 500 g particle that is released from rest at A. What are the particle's speeds at B, C, and D?

FIGURE EX10.28 shows the potential-energy diagram for a 500 g particle as it moves along the x-axis. Suppose the particle's mechanical energy is 12 J. Where are the particle's turning points?