An engineer wants to find the rotational inertia of an oddshaped object of mass \(11.3 \mathrm{~kg}\) about an axis through its center of mass. The object is supported with a wire through its center of mass and along the desired axis. The wire has a torsional constant \(\kappa=0.513 \mathrm{~N} \cdot \mathrm{m} .\) The engineer observes that this torsional pendulum oscillates through \(20.0\) cycles in \(48.7 \mathrm{~s}\). What value of the rotational inertia is calculated?

Short Answer

Expert verified
The calculated value for the rotational inertia is \(0.1586 kg \cdot m^2\)

Step by step solution

01

Calculate Period of Oscillation

The first step is to calculate the period of single oscillation. The time for 20 cycles is given as 48.7s. To find the time for a single oscillation, we divide the total time by the number of cycles: \(T = \frac{48.7s}{20} = 2.435s\)
02

Derive equation for I

The next step is to rearrange the formula for the period of oscillation to solve for I (moment of inertia). \(T = 2\pi\sqrt{\frac{I}{\kappa}}\) Squaring both sides, we get \(T^2 = (2\pi)^2\frac{I}{\kappa}\). Rearranging to solve for I gives us \(I = \frac{T^2\kappa}{(2\pi)^2}\)
03

Substitute values

Finally, substituting the value of \(T = 2.435s\) and \(\kappa = 0.513N \cdot m\) into the equation from step 2, we get \(I = \frac{(2.435s)^2 * 0.513N \cdot m}{(2\pi)^2} = 0.1586 kg \cdot m^2\)

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

Starting from Eq. \(17-43\), find the velocity \(v_{x}(=d x / d t)\) in forced oscillatory motion. Show that the velocity amplitude is $$ v_{\mathrm{m}}=F_{\mathrm{m}} /\left[\left(m \omega^{\prime \prime}-k / \omega^{\prime}\right)^{2}+b^{2}\right]^{1 / 2} $$ The equations of Section \(17-8\) are identical in form with those representing an electrical circuit containing a resistance \(R\), and inductance \(L\), and a capacitance \(C\) in series with an alternating emf \(V=V_{\mathrm{m}} \cos \omega^{\prime \prime} t\). Hence \(b, m, k\), and \(F_{\mathrm{m}}\), are analogous to \(R, L, 1 / C\), and \(V_{\mathrm{m}}\), respectively, and \(x\) and \(v\) are analogous to electric charge \(q\) and current \(i\), respectively. In the electrical case the current amplitude \(i_{\mathrm{m}}\), analogous to the velocity amplitude \(v_{\mathrm{m}}\) above, is used to describe the quality of the resonance.

(a) Calculate the reduced mass of each of the following diatomic molecules: \(\mathrm{O}_{2}, \mathrm{HF}\), and \(\mathrm{CO}\). Express your answers in unified atomic mass units, the mass of a hydrogen atom being \(1.01\) u. (b) An HF molecule is known to vibrate at a frequency of \(f=8.7 \times 10^{13} \mathrm{~Hz} .\) Find the effective "force constant" \(k\) for the coupling forces between the atoms. In terms of your experience with ordinary springs, would you say that this "molecular spring" is relatively stiff or not?

An oscillator consists of a block of mass \(512 \mathrm{~g}\) connected to a spring. When set into oscillation with amplitude \(34.7 \mathrm{~cm}\), it is observed to repeat its motion every \(0.484 \mathrm{~s}\). Find \((a)\) the period, \((b)\) the frquency, \((c)\) the angular frequency, \((d)\) the force constant, \((e)\) the maximum speed, and \((f)\) the maximum force exerted on the block.

A body oscillates with simple harmonic motion according to the equation $$ x=(6.12 \mathrm{~m}) \cos [(8.38 \mathrm{rad} / \mathrm{s}) t+1.92 \mathrm{rad}] $$ Find \((a)\) the displacement, \((b)\) the velocity, and \((c)\) the acceleration at the time \(t=1.90 \mathrm{~s}\). Find also \((d)\) the frequency and (e) the period of the motion. The scale of a snrine balance reading from 0 to \(500 \mathrm{lh}\) is

An automobile can be considered to be mounted on four springs as far as vertical oscillations are concerned. The springs of a certain car of mass \(1460 \mathrm{~kg}\) are adjusted so that the vibrations have a frequency of \(2.95 \mathrm{~Hz}\). (a) Find the force constant of each of the four springs (assumed identical). (b) What will be the vibration frequency if five persons, averaging \(73.2\) kg each, ride in the car?

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