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Chapter 13: Problem 63

An object of mass \(2.00 \mathrm{~kg}\) is oscillating freely on a vertical spring with a period of \(0.600 \mathrm{~s}\). Another object of unknown mass on the same spring oscillates with a period of \(1.05 \mathrm{~s}\). Find (a) the spring constant \(k\) and (b) the unknown mass.

Short Answer

Expert verified
The spring constant \(k\) can be obtained by substituting corresponding values into the formula \(k = \frac{4\pi^2m}{T^2}\). The unknown mass \(m_2\) can be found by substituting the obtained \(k\) and the given \(T\) into the formula \(m_2 = \frac{kT^2}{4\pi^2}\). Calculate the values to get the final results.

Step by step solution

01

Determine the spring constant \(k\) using the first object

The formula for the period \(T\) of an object of mass \(m\) oscillating on a spring with spring constant \(k\) is given by \(T = 2\pi \sqrt{\frac{m}{k}}\). For the first object, we're given that \(T = 0.600s\) and \(m = 2.00kg\). We can rearrange this formula to solve for \(k\): \(k = \frac{4\pi^2m}{T^2}\).
02

Calculate the spring constant \(k\)

Given values of \(m = 2.00kg\) and \(T = 0.600s\), we can substitute these values into the derived formula: \(k = \frac{4\pi^2(2.00)}{(0.600)^2}\). This will give us a numerical value for \(k\).
03

Determine the unknown mass using the second object

The period \(T\) for the second object with the unknown mass \(m_2\) is given by \(T = 1.05s\). Using the same formula as before, but this time rearranging it to solve for \(m_2\), we get: \(m_2 = \frac{kT^2}{4\pi^2}\).
04

Calculate the unknown mass

Now, substitute the values \(k\) (obtained in Step 2) and \(T = 1.05s\) in the formula: \(m_2 = \frac{k(1.05)^2}{4\pi^2}\). This will give us the value of unknown mass \(m_2\).

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Most popular questions from this chapter

A simple pendulum has a length of \(52.0 \mathrm{~cm}\) and makes \(82.0\) complete oscillations in \(2.00 \mathrm{~min}\). Find (a) the period of the pendulum and (b) the value of \(g\) at the location of the pendulum.

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