A particle of mass m moves in a straight line under the action of a conservative...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = x² e-x (1) Calculate F(x) and find the two equilibrium points of the system (i.e., points xe such that F(x) = 0). Compute if the equilibria are stable (i.e., local minima of the potential energy: U"(x) > 0) or unstable (i.e., local maxima of the potential energy: U"(x) < 0). Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system (i.e., kinetic plus potential), in terms of v and x. Show that dE de = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion m dv dr = F) (iii) Assume the particle starts in xp =0 with positive initial velocity vo > 0. Find the initial energy Eg of the particle. Using (ii), show that the particle reaches x = 2 only if vo>v', with v' = Be and in this case the particle's velocity in x = 2 is v(2) = 8e-2 F 26 + 213 A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = x² e-x (1) Calculate F(x) and find the two equilibrium points of the system (i.e., points xe such that F(x) = 0). Compute if the equilibria are stable (i.e., local minima of the potential energy: U"(x) > 0) or unstable (i.e., local maxima of the potential energy: U"(x) < 0). Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system (i.e., kinetic plus potential), in terms of v and x. Show that dE de = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion m dv dr = F) (iii) Assume the particle starts in xp =0 with positive initial velocity vo > 0. Find the initial energy Eg of the particle. Using (ii), show that the particle reaches x = 2 only if vo>v', with v' = Be and in this case the particle's velocity in x = 2 is v(2) = 8e-2 F 26 + 213
Expert Answer:
Answer rating: 100% (QA)
To solve the given problem well go through each part step by step i Calculate Fx and find the equilibrium points To find the force Fx we can take the ... View the full answer
Related Book For
Fundamentals of Physics
ISBN: 978-0471758013
8th Extended edition
Authors: Jearl Walker, Halliday Resnick
Posted Date:
Students also viewed these mathematics questions
-
The net force on a particle of mass m has the potential energy function graphed in Fig. 7.24a. If the total energy is E 1 , graph the speed v of the particle versus its position x. At what value of x...
-
A particle of mass m moves in the plane xy due to the force varying with velocity as F = a (yi = xj), where a is a positive constant, i and j are the unit vectors of the x and y axes. At the initial...
-
A particular paper included the accompanying data on the tar level of cigarettes smoked for a sample of male smokers who subsequently died of lung cancer. Assume it is reasonable to regard the sample...
-
Which Of the following, which is the least likely component of the project charter? This type of question contains radio buttons and checkboxes for selection of options. Use Tab for navigation and...
-
When examining the suspended gamboge droplets under a microscope, their average numbers in the layers separated by the distance h = 40 m were found to differ by = 2.0 times. The environmental...
-
If a 6.13-g sample of an isotope having a mass number of 124 decays at a rate of 0.350Ci, what is its half-life?
-
Identify the main users of financial information and discuss their needs.
-
A monopolists inverse demand function is P = 100 Q. The company produces output at two facilities; the marginal cost of producing at facility 1 is MC1(Q1) = 4Q1, and the marginal cost of producing...
-
Hull Baking Company uses 250,000 lb of sugar annually in its cakes and pastries. The company can order the sugar in 100 lb bags, at the cost of $19.50 per bag, delivered. The cost of preparing and...
-
Hastings Company bases its variable overhead performance report on the actual direct labour- hours of the period. Data concerning the most recent year, which ended on December 31, are as follows:...
-
You are a marine biologist and are interested in why some Black Turban Snails have a shell that spins to the left and some that spin to the right. Through some research, you find that which direction...
-
A component has a mean time between failure of 1500 hours, and a mean time to repair of 30 hours. What is the availability of this component?
-
Describe three (3) different ways to 'frame' organizational messaging .
-
Sales for Gonzalez Pro's Sports Equipment are expected to be 61,000 units for the coming month. The company likes to maintain 10 percent of unit sales for each month in ending inventory. Beginning...
-
Sally has $18,960 in her investment account. She buys 2,000 shares of DEF Inc. for $12.00 per share. The margin requirement for DEF is 30%. a. How much in excess of the minimum requirement did Sally...
-
Outdoor Luggage, Incorporated, makes high-end hard-sided luggage for sports equipment. Data concerning three of the company's most popular models appear below: Ski Guard Selling price per unit $ 300...
-
These items are taken from the financial statements of Ivanhoe Company at December 31, 2022 Buildings Accounts receivable: Prepaid insurance Cash Equipment Land Insurance expense Depreciation expense...
-
For each equation, (a) Write it in slope-intercept form (b) Give the slope of the line (c) Give the y-intercept (d) Graph the line. 7x - 3y = 3
-
(a) In a double-slit experiment, what ratio of d to a causes diffraction to eliminate the fourth bright side fringe? (b) What other bright fringes are also eliminated?
-
If the gauge number of a wire is increased by 6, the diameter is halved; if a gauge number is increased by 1, the diameter decreases by the factor 21/6 (see the table in Problem 6). Knowing this, and...
-
In Fig. 4-35, particle A moves along the line y = 30 m with a constant velocity v of magnitude 3.0 m/s and parallel to the x axis. At the instant particle A passes the y axis, particle B leaves the...
-
Which type of business organization is owned by its stockholders? a. Proprietorship b. Partnership C. Corporation d. All the above are owned by stockholders.
-
Consider the overall effects of transactions in questions 5, 6, and 7 on Boardmaster. What is Boardmasters net income or net loss? a. Net income of \($57,000 b. \) Net loss of \($3,000\) C. Net...
-
The income statement reports a. financial position on a specific date. b. results of operations on a specific date. C. financial position for a specific period. d. results of operations for a...
Study smarter with the SolutionInn App