Two packages are placed on a spring scale whose plate weighs \(10 \mathrm{lb}\) and whose stiffness is \(50 \mathrm{lb} / \mathrm{in}\). When one package is accidentally knocked off the scale the remaining package is observed to oscillate through 3 cycles per second. What is the weight of the remaining package?

Short Answer

Expert verified
Weight of remaining package ≈ 27.24 lb.

Step by step solution

01

Convert the frequency to angular frequency

To find the angular frequency, we can use the following formula: \(\omega = 2\pi f\) Substitute the given frequency value, \(f = 3\,\text{cycles/s}\): \(\omega = 2\pi (3) = 6\pi\,\text{rad/s}\)
02

Use frequency formula to find mass

With the angular frequency found in step 1, we can use the formula for the frequency of a spring-mass system to find the mass of the remaining package. First, we need to rearrange the frequency formula to solve for mass, \(m\). Frequency Formula: \(f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\) After rearranging, we get \(m=\frac{k}{(2\pi f)^2}\) Using the given stiffness (lb/in) \(k=50\) and the angular frequency (rad/s) \(\omega = 6\pi\), we can find the mass (in slugs) as follows: \(m=\frac{50}{(6\pi)^2}\)
03

Convert mass to weight

With mass, \(m\), in slugs, we can convert it to weight. To find the weight, we use the formula, Weight = Mass \(\times\) Gravity, where gravity is \(32.2\,\text{ft/s}^2\), and 1 slug is equivalent to 32.174 lb. Plug in the mass, \(m\), and the gravity constant, \(g=32.2\,\text{ft/s}^2\), to get the weight: \(W = (50/(6\pi)^2) \times 32.2\)
04

Subtract the weight of the spring scale's plate

Finally, subtract the weight of the spring scale's plate (10 lb) from the total weight found in step 3 to determine the weight of the remaining package: Weight of remaining package = \(((50/(6\pi)^2) \times 32.2) - 10\) Evaluate the expression and round your answer: Weight of remaining package = \(#Weight#\) The remaining package weighs approximately #Weight# lb.

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Most popular questions from this chapter

The cranking device shown consists of a mass-spring system of stiffiness \(k\) and mass \(m\) that is pin-connected to a massless rod which, in turn, is pin- connected to a wheel at radius \(R\), as indicated. If the mass moment of inertia of the wheel about an axis through the hub is \(I_{O}\), determine the natural frequency of the system. (The spring is unstretched when connecting pin is directly over hub ' \(O\) '.)

A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\) and a viscous damper whose coefficient is \(2 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). (a) Determine the response of the horizontally configured system if the mass is displaced 1 meter to the right and released from rest. Plot and label the response history of the system. (b) Determine the response and plot its history if the damping coefficient is \(8 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\).

Determine the natural period of a typical ice cube floating in water. Measure the dimensions of a typical cube from your refrigerator and calculate its natural frequency in water. (The dimensions may vary depending on your particular ice tray.) Confirm your "experiment." Place an ice cube in water, displace it slightly and release it. Make an approximate measure of the period of an oscillation with your wrist watch, or a stop watch if available. Repeat this operation several times and compare the average measured value with the calculated value.

Determine the overshoot of the system of Problem \(2.25\) if it is critically damped and \(v_{0}=-4 \mathrm{~m} / \mathrm{sec}\).

A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\). Determine the response of the vertically configured system if the mass is displaced 1 meter downward and released from rest. What is the amplitude, period and phase lag for the motion? Sketch and label the response history of the system.

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