A 250gblock is dropped onto a relaxed vertical spring that has a spring constant of K=2.5N/cm(Fig.).The block becomes attached to the spring and compresses the spring12cmbefore momentarily stopping. While the spring is being compressed, what work is done on the block by (a) the gravitational force on it and (b) the spring force? (c) What is the speed of the block just before it hits the spring? (Assume that friction is negligible.) (d) If the speed at impact is doubled, what is the maximum compression of the spring?

Short Answer

Expert verified
  1. Work done by gravitational force is0.29J
  1. Work done by the spring is-1.8J
  2. Speed of the object before it hit the spring is3.5m/s
  3. If Vf=2Vf, then distance of compression is0.23m

Step by step solution

01

Given information

Mass of the block ism=250kg

Spring constant isk=2.5N/cm=250N/m

Compressed distance isd=12cm=0.12m

02

Determining the concept

This problem deals with the work-energy theorem which states that the net work done by the forces applied on object is equal to the change in its kinetic energy. To solve this problem, use the work-energy theorem and the equation relating work done and force.

Formulae:

W=FxcosϕW=K

Where,

F is force, xis displacement,K is change in kinetic energy and Wis the work done.

03

(a) Determining the work done by gravitational force ()

Wg=Fgdcosϕϕ=0°mg=mgdcos0mg=0.25×9.8×0.12mg=0.294J0.29J (i)

Hence, the work done by gravitational force is Wg=0.29J

04

(b) determining the work done by the spring (Ws)

Kinetic energy of the spring is

K=12kd2Ws=K=Ki-Kf

As the initial position of the spring is mean position,

role="math" localid="1657176614801" Ws=-KfWs=-12kd2(ii)Ws=-12×250×(0.12)2Ws=-1.8J

Hence, the work done by the spring isWs=-1.8J

05

(c) Determining the speed of the object before it hit the spring ( v )

Wnet=K=Ki-Kf

Astheinitial position of the spring is mean position,Ki=0,

Wnet=-KfWs+Wg=-12mvf2-1.8+0.294=-12×250×vf2vf2=2×1.5060.25vf=3.471m/svf=3.5m/s

Hence, the speed of the object before it hit the springv=3.5m/s.

06

(d) Determining the distance of compression d if Vf=2Vf

Wnet=-KfWs+Wg=-12mvf2Usingvf=2vfandequation(iand2,-12kd2+mgd=-12m(2vf)2 Solvingandrearranging,d2-2mgdk-mvf2k=0Thisisthequardraticequationwith,a=1,b=-2mgk,c=-mvf2kAsthenegativevaluewillgiveanegativeroot,consideronlythepositivefactor,d=-b+b2-4ac2ad=--2mgk+-2mgk2-4-mvf2k2ad=2mgk+2k4m2g2k2+4mvf2k2d=2mgk+2km2k2+mvf2k2d=mg+m2g2+mvf2kkd=0.25×9.8+0.252(9.8)2+0.25(3.471)2250250Bysolvingtheequation,d=0.23m

Hence, ifv'=2v, then distance of compression is d=0.23m.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Is positive or negative work done by a constant force fon a particle during a straight-line displacement dif (a) the angle between fanddis30°; (b) the angle is 100°; (c) localid="1657168288688" f=2i^-3j^and d=-4i^?

The only force acting on a2.0kgbody as it moves along a positive x axis has an x component, fx=-6xNwith x in meters. The velocity atx=3.0mis8.0m/s. (a) What is the velocity of the body atx=4.0m? (b) At what positive value of x will the body have a velocity of5.0m/s?

On August 10, 1972, a large meteorite skipped across the atmosphere above the western United States and western Canada, much like a stone skipped across water. The accompanying fireball was so bright that it could be seen in the daytime sky and was brighter than the usual meteorite trail. The meteorite’s mass was about 4×106kg; its speed was about 15km/s. Had it entered the atmosphere vertically, it would have hit Earth’s surface with about the same speed. (a) Calculate the meteorite’s loss of kinetic energy (in joules) that would have been associated with the vertical impact. (b) Express the energy as a multiple of the explosive energy of 1 megaton of TNT, which is 4.2×1015J. (c) The energy associated with the atomic bomb explosion over Hiroshima was equivalent to 13kilotons of TNT. To how many Hiroshima bombs would the meteorite impact have been equivalent?

A 0.30 kgladle sliding on a horizontal frictionless surface is attached to one end of a horizontal spring (k=500N/m) whose other end is fixed. The ladle has a kinetic energy of 10 J as it passes through its equilibrium position (the point at which the spring force is zero). (a) At what rate is the spring doing work on the ladle as the ladle passes through its equilibrium position? (b) At what rate is the spring doing work on the ladle when the spring is compressed 0.10 mand the ladle is moving away from the equilibrium position?

Spring A is stiffer than spring B (kA>kB). The spring force of which spring does more work if the springs are compressed, (a) the same distance and (b) by the same applied force?

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free