Find extension of new equilibrium position by suspended mass

Assignment Help Physics
Reference no: EM13960167

1) Evaluate the work done W= Int (from O to P) F  dr = Int (from O to P) (Fx dx + Fy dy) by the two-dimensional force F = (x2, 2x, y) along the three paths joining the origin to the point P = (1, 1) and defined as follows: (a) This path goes along the x axis to Q = (1,0) and then straight up to P. (Divide the integral into two pieces, Int (from O to P) = Int (from O to Q) + Int (from P to Q). (b) On this path y = x2, and you can replace the term dy in W by dy = 2xdx and convert the whole integral into an integral over x. (c) This path is given parametrically as x = t3, y = t2. In this case rewrite x, y, dx and dy in W and convert the integral into an integral over t.

1853_curve.png

2) A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string, whose other end passes through a hole in the table, where I am holding it. Initially the particle is moving in a circle of radius ro with angular velocity wo, but I now pull the string down through the hole until a length r remains between the hole and the particle. (b) Assuming that I pull the string so slowly that we can approximate the particle's path by a circle of slowly shrinking radius, calculate the work I did pulling the string. (b) Compare your answer to part a with the particle's gain in kinetic energy.

3) (a) Consider a mass m in a uniform gravitational field g, so that the force on m is mg, where g is a show that the work done by gravity is Wgrav(1 τ 2) = -mgh where h is the vertical height gained between points 1 and 2. Use this result to prove that the force of gravity is conservative (at least in a region small enough so that g can be considered constant).

(b) Show that, if we choose axes with y measured vertically up, the gravitational potential energy is U = mgy (if we choose U = 0 at the origin).

4) For a system of N particles subject to a uniform gravitational field g acting vertically down, prove that the total gravitational energy is the same as if all the mass were concentrated at the CM of the system, that is: U = Summation (alpha on bottom of summation) Ualpha = MgY, where M = summation of Malpha is the total mass and R = (X, Y, Z) is the position of the CM, with the y coordinate measured vertically up.

5) Consider a small frictionless puck perched at the top of a fixed sphere of radius R. If the pluck is given a tiny nudge so that it begins to slide down, through what vertical height will it descend before it leaves the surface of the sphere? [Hint: Use conservation of energy and Newton's second law.]

6) (a) The force exerted by a one-dimensional spring, fixed at one end, is F = -kx, where x is the displacement of the other end from its equilibrium position. Assuming that this force is conservative (which it is) show that the corresponding potential energy is U = (1/2) kx2, if we choose U to be zero at the equilibrium position.

(b) Suppose that this spring is vertically hung from the ceiling with a mass m suspended from the other end and constrained to move in the vertical direction only. Find the extension xo of the new equilibrium position with the suspended mass. Show that the total potential energy (spring plus gravity) has the same form (1/2) ky2 if we use coordinate y equal to the displacement measured from the new equilibrium position at x = xo (and redefine our reference point so that U = 0 at y = 0).

Reference no: EM13960167

Questions Cloud

Find the marginal distributions of x and of y : The manager accepts 90 bookings. What is the probability that more than 50 clients turn up?
Expected time and variance for activity : a. Use PERT to identify the expected time and variance for each activity. b. What is the critical path? c. What is the expected time and variance of the critical path?
Problems of real estate development : They could bring a lawsuit, but they are afraid the case is so complicated that a judge and jury not familiar with the problems of real estate development would not reach a proper result. What can they do?
Allocate the service department costs : Allocate the service department costs to production departments using the direct method. Round to two decimal places.
Find extension of new equilibrium position by suspended mass : Consider a small frictionless puck perched at the top of a fixed sphere of radius R. If the pluck is given a tiny nudge so that it begins to slide down, through what vertical height will it descend before it leaves the surface of the sphere?
Write about sexual assault victims : Select a topic in Psychology that you would like to investigate further. You can choose a topic from the list provided, or let the text/companion site inspire you. write about sexual assault victims
Cost change control system : Your new project is coming in over budget and requires a cost change through the cost change control system. The cost variance was projected using which of the following tools and techniques?
Will companys strategy to market new products be successful : Chobani Inc. is venturing beyond the basic yogurt cups that made it famous. In your view, will the company's strategy to market the above new products be successful? Why or Why not?
A runner wants to run 14.4 km : A runner wants to run 14.4 km. She knows that her running pace is 7.2 mi/h. How many minutes must she run? Hint: Use 7.2 mi/h as a conversion factor between distance and time. (If you could please show steps of how you got the answer) Thank you

Reviews

Write a Review

Physics Questions & Answers

  Find the magnitude of the resulting magnetic field

A sphere of radius R is uniformly charged to a total charge of Q. It is made to spin about an axis that passes through its center with an angular speed ω. Find the magnitude of the resulting magnetic field at the center of the sphere.

  Find the equivalent resistance

A resistor is in the shape of a cube, with each side of resistance  R . Find the equivalent resistance between any two of its adjacent corners.

  What is the electric field at the location

Question: Field and force with three charges? What is the electric field at the location of Q1, due to  Q 2 ?

  What is the maximum displacement of the bridge deck

What is the maximum displacement of the bridge deck?

  What is the magnitude of the current in the wire

What is the magnitude of the current in the wire as a function of time?

  Blackbody

Questions on blackbody, Infra-Red Detectors & Optic Lens and Digital Image.

  Gravity conveyor

Illustrate the cause of the components accelerating from rest down the conveyor.

  Calculate the dc voltage

Calculate the dc voltage applied to the circuit.

  Quadrupole moments in the shell model

Quadrupole moments in the shell model

  Determine the tension in each string

Determine the tension in each string

  Introductory mechanics: dynamics

Calculate the smallest coefficient of static friction necessary for mass A to remain stationary.

  Evaluate maximum altitude

Evaluate maximum altitude?

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd