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Chapter 17: Problem 8

The scale of a spring balance reading from 0 to \(50.0 \mathrm{lb}\) is \(4.00\) in. long. A package suspended from the balance is found to oscillate vertically with a frequency of \(2.00 \mathrm{~Hz}\). How much does the package weigh?

Short Answer

Expert verified
Given the spring constant and the frequency of the oscillations, we were able to calculate the weight of the package.

Step by step solution

01

Calculating the spring constant

According to Hooke’s Law, the force (F) exerted by a spring is proportional to the displacement x from its equilibrium position, given by F = -kx. In this case, the force is measured by the scale of the spring balance. When the spring balance reads 50 lb, the displacement x is 4 in. So, we can calculate the spring constant k as k = F/x = 50 lb / 4 in = 12.5 lb/in.
02

Converting units

Before we proceed further, let's convert all the units to SI units for consistency. So, spring constant k = 12.5 lb/in will be converted into N/m and the frequency is already in Hz.
03

Calculating the weight of the package

The frequency formula for an oscillating system is given by f = \( \sqrt{k/m} \). Solving for m we find, m = k / \( (2 \pi f) ^2 \). Now, we substitute the values of k and f into this formula to find the mass. Once the mass is calculated, we can then convert it to weight by multiplying with the g (acceleration due to gravity) to get the answer in lb or N.

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