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Chapter 10: Q. 62 (page 259)

A particle that can move along the x-axis experiences an interaction force $F_{x} = (3 x^{2} - 5 x) N,$ where x is inm. Find an expression for the system’s potential energy.

Short Answer

Expert verified

Expression for potential energy is $U (x) = \frac{5 x^{2}}{2} - x^{3}$

Step by step solution

01

Given information

The interactive force is given as

$F_{x} = (3 x^{2} - 5 x) N$

02

Explanation

$F_{x} = (3 x^{2} - 5 x) N U (x) = - \int F_{x} dx = - \int (3 x^{2} - 5 x) dx = - \int 3 x^{2} dx + \int 5 xdx = - 3 [\frac{x^{3}}{3}] + 5 [\frac{x^{2}}{2}] U (x) = - x^{3} + \frac{5 x^{2}}{2} U (x) = \frac{5 x^{2}}{2} - x^{3}$

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Most popular questions from this chapter

The lowest point in Death Valley is $85 m$ below sea level. The summit of nearby Mt. Whitney has an elevation of $4420 m$ . What is the change in potential energy when an energetic $65 k g$ hiker makes it from the floor of Death Valley to the top of Mt. Whitney?

$F I G U R E E X 10.34$ shows the potential energy of a system in which a particle moves along the x-axis. Draw a graph of the force $F_{x}$ as a function of position x.

A freight company uses a compressed spring to shoot $2.0 k g$ packages up a -high frictionless ramp into a truck, as FIGURE $P 10.51$ shows. The spring constant is $500 N / m$ and the spring is compressed $30 c m$ .

a. What is the speed of the package when it reaches the truck?

b. A careless worker spills his soda on the ramp. This creates a $50 - c m$ -long sticky spot with a coefficient of kinetic friction $0.30$ . Will the next package make it into the truck?

It’s your birthday, and to celebrate you’re going to make your first bungee jump. You stand on a bridge $100 m$ above a raging river and attach a $30 m$ -long bungee cord to your harness. A bungee cord, for practical purposes, is just a long spring, and this cord has a spring constant of $40 N / m$ . Assume that your mass is $80 k g$ . After a long hesitation, you dive off the bridge. How far are you above the water when the cord reaches its maximum elongation?

Rank in order, from most to least, the elastic potential energy ${(U_{s_{p}})}_{a} t o {(U_{S_{p}})}_{d}$ stored in the springs of FIGURE $Q 10.5$ Explain.

See all solutions

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