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Chapter 17: Problem 13

An oscillator consists of a block attached to a spring \((k=\) \(456 \mathrm{~N} / \mathrm{m}) .\) At some time \(t\), the position (measured from the equilibrium location), velocity, and acceleration of the block are \(x=0.112 \mathrm{~m}, v_{x}=-13.6 \mathrm{~m} / \mathrm{s}, a_{x}=-123 \mathrm{~m} / \mathrm{s}^{2} .\) Calcu- late \((a)\) the frequency, \((b)\) the mass of the block, and \((c)\) the amplitude of oscillation.

Short Answer

Expert verified
The frequency of the oscillator is approximately \(\frac{1}{2\pi}\sqrt{\frac{456}{123}}\) Hz, the mass of the block is \(\frac{456}{123}\) kg, and the amplitude of the oscillation is roughly \(\sqrt{(0.112)^2 + (\frac{-13.6}{2\pi f})^2}\) m.

Step by step solution

01

Find the Frequency

First, calculate the frequency of the system using the formula \(f = \frac{1}{2\pi}\sqrt{\frac{k}{a_x}}\). Substituting the given values \(k = 456 \mathrm{~N/m}\) and \(a_x = -123 \mathrm{~m/s}^2\).Note, you will take the absolute value of \(a_x\) as acceleration can never be negative.The frequency \(f\) is therefore \(\frac{1}{2\pi}\sqrt{\frac{456}{123}}\).
02

Calculate the Mass

Next, find the mass of the block using the formula \(m = \frac{k}{a_x}\). Substitute the given values to get the mass \(m = \frac{456}{123}\).
03

Determine the Amplitude

Finally, calculate the amplitude of the oscillation using the formula \(A = \sqrt{x^2 + (\frac{v_x}{2\pi f})^2}\). Substitute the calculated frequency and given values \(x = 0.112 \mathrm{~m}\) and \(v_x = -13.6\mathrm{~m/s}\) (take absolute value) to find the amplitude \(A = \sqrt{(0.112)^2 + (\frac{-13.6}{2\pi f})^2}\).

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Most popular questions from this chapter

A \(5.22-\mathrm{kg}\) object is attached to the bottom of a vertical spring and set vibrating. The maximum speed of the object is \(15.3 \mathrm{~cm} / \mathrm{s}\) and the period is \(645 \mathrm{~ms}\). Find \((a)\) the force constant of the spring, (b) the amplitude of the motion, and \((c)\) the frequency of oscillation.

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?

A simple pendulum of length \(1.53 \mathrm{~m}\) makes \(72.0\) oscillations in \(180 \mathrm{~s}\) at a certain location. Find the acceleration due to gravity at this point.

A \(95.2-\mathrm{kg}\) solid sphere with a \(14.8-\mathrm{cm}\) radius is suspended by a vertical wire attached to the ceiling of a room. A torque of \(0.192 \mathrm{~N} \cdot \mathrm{m}\) is required to swist the sphere through an angle of \(0.850 \mathrm{rad} .\) Find the period of oscillation when the sphere is released from this position.

A damped harmonic oscillator involves a block (m = \(1.91 \mathrm{~kg}\) ), a spring \((k=12.6 \mathrm{~N} / \mathrm{m})\), and a damping force \(F=\) \(-b v_{x} .\) Initially, it oscillates with an amplitude of \(26.2 \mathrm{~cm}\); because of the damping, the amplitude falls to three-fourths of this initial value after four complete cycles. (a) What is the value of \(b ?(b)\) How much energy has been "lost" during these four cycles?
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