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Chapter 2: Problem 2

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
Answer: The approximate weight of the remaining package is 5.66 lb.

Step by step solution

01

Determine the combined weight of the packages on the scale

First, let W1 and W2 be the weight of the first and second package, respectively. The combined weight of both packages is given by W1 + W2. The spring with both packages will compress by an amount x, such that W1 + W2 = kx, where k is the stiffness of the spring (50 lb/in).
02

Determine the combined mass of the packages with the given weight

We know that weight = mass × gravity. Thus, the mass M1 and M2 of the first and second package can be calculated as M1 = W1/32.2 ft/s² and M2 = W2/32.2 ft/s² (Assuming standard gravity is ~32.2 ft/s²).
03

Obtain the total mass on the spring

When both packages are on the spring, the total mass on the spring is M = M1 + M2.
04

Calculate the spring's natural frequency

We are given that the natural frequency f of the remaining package after knocking off one of the packages is 3 cycles per second. Using the formula for the frequency of oscillation under spring mass system, f = (1 / (2 * pi)) * sqrt(k/M), where k is the spring stiffness (50 lb/in) and M is the mass of the remaining package.
05

Solve for the weight of the remaining package, W2

First, we need to rearrange the formula for frequency in terms of M: $$ M = \frac{k}{{(2\cdot\pi\cdot f)}^2} $$From the previous steps we know W1+W2= kx and the mass of remaining package is M2=W2/32.2 = $$ \frac{k}{{(2\cdot\pi\cdot f)}^2} $$. Now, multiplying both sides by 32.2, we obtain: $$ W2 = 32.2\cdot\frac{k}{{(2\cdot\pi\cdot f)}^2} $$Substitute the given values for k, f, and 32.2 ft/s² for gravity, and solve for W2.
06

Calculate the weight of the remaining package

Finally, by substituting the given values (k=50 lb/in and f=3 cycles/s) into the equation, we get: $$ W2 = 32.2\cdot\frac{50}{(2\cdot(3.1416)\cdot 3)^2} $$After evaluating the expression, you get W2 to be approximately 5.66 lb. Therefore, the weight of the remaining package is approximately 5.66 lb.

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Most popular questions from this chapter

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 \(8 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\).

A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\). If the coefficients of static and kinetic friction between the block and the surface it moves on are respectively \(\mu_{s}=\) \(0.12\) and \(\mu_{k}=0.10\), determine the drop in amplitude between successive periods during free vibration. What is the frequency of the oscillations?

The system shown consists of a rigid rod, a flywheel of radius \(R\) and mass \(m\), and an elastic belt of stiffness \(k\). Determine the natural frequency of the system. The belt is unstretched when \(\theta=0\).

The cranking device shown consists of a mass-spring system of stiffness \(k\) 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 \(30 \mathrm{~cm}\) aluminum rod possessing a circular cross section of \(1.25 \mathrm{~cm}\) radius is inserted into a testing machine where it is fixed at one end and attached to a load cell at the other end. At some point during a tensile test the clamp at the load cell slips, releasing that end of the rod. If the \(20 \mathrm{~kg}\) clamp remains attached to the end of the rod, determine the frequency of the oscillations of the rod-clamp system?
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