Chapter 7, Problem 7.2P

As an illustration of why it matters which variables you hold fixed when taking partial derivatives, consider the following mathematical example. Let w = xy and x = yz. Write w purely in terms of x and z, and then purely in terms of y and z.

Note: Because the argument of the trigonometric functions in this problem will be unitless, your calculator must be in radian mode if you use it to evaluate any trigonometric functions. You will likely need to switch your calculator back into degree mode after this problem.A massless spring is hanging vertically. With no load on the spring, it has a length of 0.24 m. When a mass of 0.59 kg is hung on it, the equilibrium length is 0.98 m. At t=0, the mass (which is at the equilibrium point) is given a velocity of 4.84 m/s downward. At t=0.32s, what is the acceleration of the mass? (Positive for upward acceleration, negative for downward)

The following problem involves an equation of the form = f(y). dy dt Sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. dy = y(y − 1)(y — 2), yo ≥0 dt The function y(t) = 0 is no equilibrium solution at all. ▼ The function y(t) = 1 is Choose one The function y(t) = 2 is Choose one

An intersting thing happens when springs systems have no attachments to the outside. Consider the following free system. m, with spring constants c 3. Assume down is the positive direction. Write the elongation matrix. A

Consider the function ƒ(x) = 4 sin(( 3)) + 6. State the amplitude A. period P 2 A, a midline. State the phase shift and vertical translation. In the full period [0, P], state the maximum a N minimum y-values and their corresponding a-values. Enter the exact answers. Amplitude: A = 4 Period: P = Bù Midline: y = Number The phase shift is Click for List The vertical translation is Click for List 1 Hints for the maximum and minimum values of f(x): π and = 2 프 The maximum value of y = sin(x) is y = 1 and the corresponding a values are x multiples of 2 7 less than and more than this a value. You may want to solve ( x − 3) = 2 3T and • The minimum value of u = sin (r) is 71 sin (r) is = -1 and the corresponding r values are Submit Assignment Quit & Save

A particle of mass m is projected upward with a velocity vo at an angle a to the horizontal in the uniform gravitational field of earth. Ignore air resistance and take the potential energy U at y=0 as 0. Using the cartesian coordinate system answer the following questions: a. find the Lagrangian in terms of x and y and identiy cyclic coordinates. b. find the conjugate momenta, identify them and discuss which are conserved and why c. using the lagrange's equations, find the x and y components of the velocity as the functions of time.

Evaluate f. A.dr from the point P(0,0, 0) to Q(1, 1, 1) along the curve r = ît + ĵt² + kr³ with x = t, y = t², z = t³, where A = yî +xzî +xyzk

Log Ride (object sliding down a circularly curved slope). In an amusement park ride, a boat moves slowly in a narrow channel of water. It then passes over a slope into a pool below as shown. The water in the channel ensures that there is very little friction. A B (R Ax On this particular ride, the slope (the black arc through points A and B) is a circular curve of radius R, centered on point P. The dotted line shows the boat's trajectory. At some point B along the slope, the boat (and the water falling with it) will separate from the track and fall freely as shown. Note that the pond is level with point P. Considering the boat as a particle, assume it starts from rest at point A and slides down the slope without friction. a) Determine the angle øsen at which the boat will separate from the track. b) Determine the horizontal distance Ax (from point P) at which the boat strikes the pond surface. c) Determine the impact speed vf and impact angle 0. Hints: Derive a formula giving the…

Answer both  Q1: An object undergoes simple harmonic motion. As itmomentarily passes through the equilibriumposition, which statement is true about its potentialenergy U and kinetic energy K?minimum U, minimum Kmaximum U, maximum Kminimum U, maximum Kmaximum U, minimum KNone of the above 2. An object undergoes simple harmonic motion. When itstops at its turnaround points, which of thefollowing statements is true about its potential energy U andkinetic energy K?minimum U, maximum Kminimum U, minimum Kmaximum U, minimum Kmaximum U, maximum KNone of the above