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Chapter 10: Problem 49

An object of mass \(306 \mathrm{g}\) is attached to the base of a spring, with spring constant \(25 \mathrm{N} / \mathrm{m},\) that is hanging from the ceiling. A pen is attached to the back of the object, so that it can write on a paper placed behind the mass-spring system. Ignore friction. (a) Describe the pattern traced on the paper if the object is held at the point where the spring is relaxed and then released at \(t=0 .\) (b) The experiment is repeated, but now the paper moves to the left at constant speed as the pen writes on it. Sketch the pattern traced on the paper. Imagine that the paper is long enough that it doesn't run out for several oscillations.

Short Answer

Expert verified
Question: Describe the pattern traced on the paper when a mass attached to a spring hanging from the ceiling undergoes simple harmonic motion, and sketch the pattern when the paper moves to the left. Answer: When the mass oscillates and the paper is stationary, the pattern traced on the paper is a straight horizontal line. When the paper moves to the left, the pattern becomes a sinusoidal wave due to the combination of the vertical oscillation and the constant horizontal motion of the paper.

Step by step solution

01

Find the time period of oscillation

To find the time period \(T\), we can use the formula \(T = 2 \pi \sqrt{\frac{m}{k}}\), where \(m\) is the mass of the object and \(k\) is the spring constant. First, convert the mass from grams to kg: \(m = 306 \, \mathrm{g} = 0.306 \, \mathrm{kg}\). Using the given values, \(m = 0.306 \, \mathrm{kg}\) and \(k = 25 \, \mathrm{N/m}\), we can calculate the time period: \(T = 2 \pi \sqrt{\frac{0.306 \, \mathrm{kg}}{25 \, \mathrm{N/m}}} = 2.206 \, \mathrm{s}\).
02

Determine the equation of motion:

Since we know the object is undergoing simple harmonic motion, we can write the equation of motion as: \(x(t) = A \sin(\omega t + \phi)\), where \(A\) is the amplitude of the oscillation, \(\omega\) is the angular frequency, \(t\) is the time, and \(\phi\) is the phase difference. The angular frequency can be found using \(\omega = \frac{2 \pi}{T}\), so \(\omega = 2.843 \, \mathrm{rad/s}\). Since the object is released from the relaxed point of the spring at \(t = 0\), the amplitude \(A\) is zero, and there is no phase difference, \(\phi = 0\). Thus, the equation of motion is \(x(t) = 0 \sin(2.843 t)\). (a)
03

Describe the pattern traced on the paper:

The object only moves vertically due to gravity and oscillates due to the spring force when released. Since the spring is initially relaxed and the amplitude is zero, the pen will trace a straight horizontal line as the object oscillates up and down. (b)
04

Sketch the pattern traced on the paper as it moves to the left:

When the paper moves to the left with a constant speed while the pen writes on it, the vertical oscillations of the spring are combined with the constant horizontal motion of the paper. The resulting pattern on the paper is a sinusoidal wave, where the peaks and troughs of the wave correspond to the displacement of the oscillating object. The horizontal component causes the entire pattern to be shifted to the left over time, creating a sinusoidal wave shape that runs along the paper.

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