An automobile can be considered to be mounted on four identical springs as far as vertical oscillations are concerned. The springs of a certain car are adjusted so that the oscillations have a frequency of 3.00 Hz.

  1. What is the spring constant of each spring if the mass of the car is 1450 kgand the mass is evenly distributed over the springs?
  2. What will be the oscillation frequency if five passengers, averaging 73.0 kgeach, ride in the car with an even distribution of mass?

Short Answer

Expert verified
  1. Spring constant is1.29×105N/m
  2. Frequency of oscillation is data-custom-editor="chemistry" 2.68Hz.

Step by step solution

01

Stating the given data

  1. Frequency of oscillations,f=3.00Hz
  2. Mass of car,mcar=1450kg
  3. Mass of each passenger, mpassenger=73kg.
02

Understanding the concept of spring constant

  1. The car is mounted on four springs. First, we calculate the mass supported by each spring as we have to calculate the spring constant. We can use the formula for angular frequency in terms of the spring constant and mass supported by each spring to find angular frequency and frequency.
  2. To calculate frequency, we first calculate the total mass by considering each passenger’s mass and the car’s mass. We can find the mass supported by each spring. We can use the spring constant found in the first part to find the frequency.

Formula:

Frequency relation to the spring constant of a body

ω=km(i)

03

a) Calculation of spring constant of each spring

The angular frequency of the motion is given as follows:

ω=2πf=2π×3=18.85rad/sec

Mass supported by each spring is one-fourth of the mass of the car. Hence, the mass load on each spring is given as

m=14504=362.5kg

Now using equation (i), the spring constant can be derived as

18.85=k362.5k=128804.41N/m=1.29×105N/m

Hence, the value of the spring constant is 1.29×105N/m.

04

b) Calculation of oscillation frequency

Now five passengers having an average mass of 73 kg ride inthecar, sothetotal mass is as follows:

M=mcar+5mpassenger=1450+573=1815kg

So, mass supported by each spring is

m=18154=453.75kg

Now, using equation (i), we get the angular frequency as follows:

ω=129000453.75=16.86rad/sec2πf=16.86rad/secangularfrequencyintermsoffrequency,ω=2πff=8.43Hz

Hence, the frequency of oscillation is 8.43Hz.

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

A simple harmonic oscillator consists of a 0.50 kgblock attached to a spring. The block slides back and forth along a straight line on a frictionless surface with equilibrium point x=0. At t=0the block is at x=0and moving in the positive x direction. A graph of the magnitude of the net forceFon the block as a function of its position is shown in Fig. 15-55. The vertical scale is set by FS=75.0N. What are (a) the amplitude and (b) the period of the motion, (c) the magnitude of the maximum acceleration, and (d) the maximum kinetic energy?

A massless spring hangs from the ceiling with a small object attached to its lower end. The object is initially held at rest in a position yisuch that the spring is at its rest length. The object is then released from yiand oscillates up and down, with its lowest position being 10cmbelowyi

(a) What is the frequency of the oscillation?

(b) What is the speed of the object when it is 8.0cmbelow the initial position?

(c) An object of mass 300gis attached to the first object, after which the system oscillates with half the original frequency. What is the mass of the first object?

(d) How far below yiis the new equilibrium (rest) position with both objects attached to the spring?

A10gparticle undergoes SHM with amplitude of 2.0mm, a maximum acceleration of magnitude8.0×103m/s2, and an unknown phase constantϕ.

(a) What is the period of the motion?

(b) What is the maximum speed of the particle?

(d) What is the total mechanical energy of the oscillator?

What is the magnitude of the force on the particle when the particle is at

(d) its maximum displacement and

(e) Half its maximum displacement?

A simple harmonic oscillator consists of a block attached to a spring with k=200 N/m. The block slides on a frictionless surface, with an equilibrium point x=0and amplitude 0.20 m. A graph of the block’s velocity v as a function of time t is shown in Fig. 15-60. The horizontal scale is set byts=0.20s. What are (a) the period of the SHM, (b) the block’s mass, (c) its displacement att=0, (d) its acceleration att=0.10s, and (e) its maximum kinetic energy.

Figure 15-24shows the x(t) curves for three experiments involving a particular spring–box system oscillating in SHM. Rank the curves according to (a) the system’s angular frequency, (b) the spring’s potential energy at time t=0, (c) the box’s kinetic energy att=0, (d) the box’s speed att=0, and (e) the box’s maximum kinetic energy, greatest first.

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