The scale of a spring balance that reads from 0to 15.0 kgis12.0cm long. A package suspended from the balance is found to oscillate vertically with a frequency of 2.00 Hz.

  1. What is the spring constant?
  2. How much does the package weigh?

Short Answer

Expert verified
  1. The spring constant is 1225 N/m.
  2. The package weighs 76.0 N.

Step by step solution

01

The given data

  • Frequency (f)=2.00 Hz.
  • Length of the spring x=12.0 cm or 0.12m.
  • Reading of scale balance ranges from m =0 or 15 kg.
02

Understanding the concept of oscillations

A particle with mass m that moves under the influence of Hooke’s law restoring force, F=-kx,exhibits simple harmonic motion. Here, Fis restoring force, kis force constant and x is the displacement from the mean position.

By using the formula for force constant k and period T, we can find the spring constant and weight of the package.

Formula:

The spring constant of the oscillations, k=F / x (i)

The period of the oscillations, T=1/f or T =2ττMk (ii)

The weight of a body, W=mg (iii)

03

a) Calculation of spring constant

Using equation (i), the spring constant of an oscillation can be given as:

k=mgxForceactingonthebody,F=mg=15kg×9.8m/s20.12m=1225N/m

Hence, the value of spring constant is 1225 N/m

04

b) Calculation of the weight of the package

The time period of a body using equation (ii) can be given as:

T=12=0.5s

Now, the mass of the package can be given using equation (ii) as:

By squaring and rearranging for M

M=T2k4ττ2=0.5s2×1225N/m43.142=7.757kg


So,theweightofthepackageusingequation(iii)canbegivenas:


W=7.757kg×9.8m/s2=76.0N


Hence,thevalueoftheweightis76.0N.

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

Although California is known for earthquakes, it has large regions dotted with precariously balanced rocks that would be easily toppled by even a mild earthquake. The rocks have stood this way for thousands of years, suggesting that major earthquakes have not occurred in those regions during that time. If an earthquake were to put such a rock into sinusoidal oscillation (parallel to the ground) with a frequency of2.2Hz, an oscillation amplitude of1.0cmwould cause the rock to topple. What would be the magnitude of the maximum acceleration of the oscillation, in terms of g?

A common device for entertaining a toddler is a jump seat that hangs from the horizontal portion of a doorframe via elastic cords (Fig. 15-63). Assume that only one cord is on each side in spite of the more realistic arrangement shown. When a child is placed in the seat, they both descend by a distance dsas the cords stretch (treat them as springs). Then the seat is pulled down an extra distance dmand released, so that the child oscillates vertically, like a block on the end of a spring. Suppose you are the safety engineer for the manufacturer of the seat. You do not want the magnitude of the child’s acceleration to exceed 0.20 gfor fear of hurting the child’s neck. If dm=10cm, what value of dscorresponds to that acceleration magnitude?

A simple pendulum of length 20 cmand mass 5.0gis suspended in a race car traveling with constant speed 70m/saround a circle of radius 50 m. If the pendulum undergoes small oscillations in a radial direction about its equilibrium position, what is the frequency of oscillation?

A 4.00kgblock hangs from a spring, extending it 16.0 cmfrom its unstretched position.

  1. What is the spring constant?
  2. The block is removed, and a0.500kgbody is hung from the same spring. If the spring is then stretched and released, what is its period of oscillation?

In Fig. 15-59, a solid cylinder attached to a horizontal spring (k=3.00 N/m) rolls without slipping along a horizontal surface. If the system is released from rest when the spring is stretched by 0.250 m , find (a) the translational kinetic energy and (b) the rotational kinetic energy of the cylinder as it passes through the equilibrium position. (c) Show that under these conditions the cylinder’s center of mass executes simple harmonic motion with period T=2π3M2k where M is the cylinder mass. (Hint: Find the time derivative of the total mechanical energy.)

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