6.4 The potential energy function for aparticle executing linear simpleharmonic motion is given by V(x) =kx /2, where k is the force constantof the oscillator. For k = 0.5 N m',V(x)the graph of V(x) versus x is shownin Fig. 6.12. Show that a particle oftotal energy 1 J moving under thispotential must 'turn back' when itreaches x = + 2 m.Fig. 6.12

force constant = K = 0.5 N/m total energy = E = 1 J

Answered: The potential energy function for a…

University Physics Volume 1

18th Edition

ISBN:9781938168277

Author:William Moebs, Samuel J. Ling, Jeff Sanny

Publisher:William Moebs, Samuel J. Ling, Jeff Sanny

Chapter8: Potential Energy And Conservation Of Energy

Section: Chapter Questions

Problem 73AP: A mysterious force acts on all particles along a particular line and always points towards a...

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A mysterious force acts on all particles along a particular line and always points towards a particular point P on the line. The magnitude of the force on a particle increases as the cube of the distance from that point; that is Fr3 , if the distance from P to the position of the particle is r. Let b be the proportionality constant, and write the magnitude of the force as F=br3. Find the potential energy of a particle subjected to this force when the particle is at a distance D from P, assuming the potential energy to be zero when the particle is at P.
Find the work done by the force F = −3y i + 4x j along one loop of the curve r = sin 2θ (in the positive direction).
A slingshot, consisting of a leather pouch attached to two rubber bands which are tied to the prongs of a wooden Y- shaped frame, is used to shoot a pebble horizontally. When the slingshot is stretched by a distance d, it gives the pebble speed v. What speed does it give the same pebble when it is stretched to distance 3d?
Consider a linear spring, as in Figure 7.7(a), with mass M uniformly distributed along its length. The left end of the spring is fixed, but the right end, at the equilibrium position x=0 , is moving with speed v in the x-direction. What is the total kinetic energy of the spring? (Hint: First express the kinetic energy of an infinitesimal element of the spring dm in terms of the total mass, equilibrium length, speed of the right-hand end, and position along the spring; then integrate.)
Consider a linear spring, as in Figure 7.7(a), with mass M uniformly distributed along its length. The left end of the spring is fixed, but the right end, at the equilibrium position x = 0, is moving with speed v in the x-direction. What is the total kinetic energy of the spring? (Hint: First express the kinetic energy of an infinitesimal element of the spring dm in terms of the total mass, equilibrium length, speed of the right-hand end, and position along the spring; then integrate.)
Given the potential energy function U(x) =−x^2 + 4x+ 2 calculate the force F(x) associated with U(x), and find the maxima/minima of the potential. Does the potential have a point of stability?
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A central force is defined to be a force that points radially,and whose magnitude and depends on only r. That is F(r)=F(r)^r.show that a central force is a conservative force,by explicity showing that∆XF=0
On an essentially frictionless, horizontal ice rink, a skater moving at v = 3.0 m/s encounters a rough patch that reduces her speed to 45% due to a friction force that is 25% of her weight (w). Use the work-energy theorem to find the length of this rough path.Express first your answer in terms of any or all of the variables v, w, and g (acceleration due togravity), and then its numerical value
Find the force corresponding to the potential energy U(x) = −a/x + b/x2.
Prove that when the average is taken with respect to position over one cycle, the average potential energy equals kA^2/6 and the average kinetic energy equals kA^2/3.
find the potential energy of a particle moving around in F(r,θ,z) = -kr^3 ˆr

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