Consider a particle of mass m moving on a smooth vertical parabolic wire placed in a...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Consider a particle of mass m moving on a smooth vertical parabolic wire placed in a uniform gravitational field: r = xE₁+yE₂, y = cx², V = mgy, where c is a positive constant. (a) Using x as a generalized coordinate, calculate the covariant basis vector along the wire. (b) Argue that the covariant component of the constraint force is zero. (e) Deduce that energy is conserved and write the energy constant E in terms of x and X. (c) Calculate the Lagrangian and obtain the second-order differential equation for x. (d) What is the equilibrium solution and the corresponding energy? (e) If the particle is started off from the equilibrium position with nonzero initial velocity, argue that a maximum height (h=) will be reached, and that the particle will subsequently oscillate forever. mg Consider a particle of mass m moving on a smooth vertical parabolic wire placed in a uniform gravitational field: r = xE₁+yE₂, y = cx², V = mgy, where c is a positive constant. (a) Using x as a generalized coordinate, calculate the covariant basis vector along the wire. (b) Argue that the covariant component of the constraint force is zero. (e) Deduce that energy is conserved and write the energy constant E in terms of x and X. (c) Calculate the Lagrangian and obtain the second-order differential equation for x. (d) What is the equilibrium solution and the corresponding energy? (e) If the particle is started off from the equilibrium position with nonzero initial velocity, argue that a maximum height (h=) will be reached, and that the particle will subsequently oscillate forever. mg
Expert Answer:
Answer rating: 100% (QA)
Solutions Step 1 a The position vector of the particle on the wire can be expressed as rxE1yE2 where E1 and E2 are the basis vectors along the x and y ... View the full answer
Related Book For
Modern Classical Physics Optics Fluids Plasmas Elasticity Relativity And Statistical Physics
ISBN: 9780691159027
1st Edition
Authors: Kip S. Thorne, Roger D. Blandford
Posted Date:
Students also viewed these mechanical engineering questions
-
How many lines per centimeter does a grating have if the third-order occurs at an 18.0o angle for 630-nm light?
-
How many slits per centimeter does a grating have if the third order occurs at a 15.0 angle for 620-nm light?
-
How many lines per centimeter must a grating have if there is to be no second-order spectrum for any visible wavelength?
-
In Exercises 8586, find a. (f g)(x); b. (g f)(x); c. (f g)(3). f(x) = x 2 + 3, g(x) = 4x - 1
-
The Dallas Armadillos, a minor-league baseball team, play their weekly games in a small stadium just outside Dallas. The stadium holds 6,000 people, and tickets sell for $20 each. The franchise owner...
-
What strategic issues have the potential to create environmental uncertainty in the following four industries: (a) Automobile; (b) Social media; (c) Newspaper; (d) Medical services?
-
Why do you think the atria of the heart are less muscular than the ventricles? Why is the left ventricle more muscular than the right ventricle?
-
Henry is a 50% partner in HJ Partnership. For 2015, the tax form he receives from HJ (Schedule K-1 of Form 1065) shows business income of $40,000. During 2015, Henry received a $10,000 distribution...
-
Define the following terms A. Interpersonal influence. B. Political influence. C. Domineering influence. D. Position influence.
-
The trial balance of Pacilio Security Services Inc. as of January 1, 2015, had the following normal balances: Cash ...................$62,860 Accounts receivable ............. 20,500 Supplies...
-
FOR YEAR5 sourcing has been just another word for procurement a financially material but strategically peripheral corporate function Now globalization aided by rapid technology innovation is changing...
-
A proton is being accelerated from rest by a uniform electric field E= 10-6N/C for 2 m. What is its final speed? Note: mass of proton is: mp=1.67 x 10-27 kg Vf=14.0 m/s Vf=384.2 m/s Vf=19.6 m/s...
-
3. A mass m is placed next to (not attached to) a spring of spring constant k on a horizontal surface, as shown. The mass is pushed a distance x to the left, compressing the spring, and released from...
-
10) Two capacitors are made with a dielectric for which x = 38.0 The capacitances with the dielectric are: C= 95.0 F and C = 60.0 F. The capacitors are charged to 100V in the circuit shown to the...
-
A horizontal force of 213 N pulls an 19 kg cabinet across a level floor. Find the acceleration of the cabinet if Uk=0.5. F
-
What horizontal force F is required to drag a 7 kg block with an acceleration of 4 m/s if a friction force of 29 N opposes the motion? -UT
-
Prepare the Petty Cash Book for the month of March: 1. Record the establishment of the Petty Cash Float in the Petty Cash Book 2. Record the Petty Cash Payment Vouchers in the Petty Cash Book 3....
-
Swifty company is a publicly held corporation whose $1 par value stock is actively traded at $30 per share. The company issued 3400 shares of stock to acquire land recently advertised at $93000. When...
-
The particle distribution function f(v, x, t) ought not to become negative if it is to remain physical. Show that this is guaranteed if it initially is everywhere nonnegative and it evolves by the...
-
Verify that the group velocity of a wave mode is perpendicular to the refractive-index surface (Fig. 21.6b). Fig. 21.6b. ck/w B (b) const @= const
-
Burkes radiation-reaction potential (27.64) produces a force per unit volume react on its nearly Newtonian source. If we multiply this force per unit volume by the velocity v = dx/dt of the sources...
-
The following are some costs incurred by a shoe manufacturer. Decide whether each one is a fixed cost or a variable cost or has some element of both. (a) The cost of leather; (b) The fee paid to an...
-
Why does the marginal cost curve pass through the bottom of the average cost curve and the average variable cost curve?
-
Outline the main factors that might influence the size of the profit mark-up set by a business.
Study smarter with the SolutionInn App