A 42.5 kg chair is attached to a spring and allowed to oscillate. When the chair is empty, it takes 1.30 s to make one complete vibration. But with a woman sitting in it, with her... A 42.5 kg chair is attached to a spring and allowed to oscillate. When the chair is empty, it takes 1.30 s to make one complete vibration. But with a woman sitting in it, with her feet off the floor, the chair now takes 2.54 s for one cycle. What is the mass of the woman? Read more

Steps to Solve

1. Identify the formula for the oscillation period

The period of oscillation $T$ for a mass-spring system is given by the formula:

$$ T = 2\pi \sqrt{\frac{m}{k}} $$

where:

• $T$ is the period
• $m$ is the mass
• $k$ is the spring constant.

2. Rearrange the formula to solve for mass

To find the mass, we can rearrange the formula:

$$ T = 2\pi \sqrt{\frac{m}{k}} $$

Squaring both sides, we get:

$$ T^2 = 4\pi^2 \frac{m}{k} $$

Now, isolating $m$, we have:

$$ m = \frac{T^2 k}{4\pi^2} $$

3. Substituting known values

If we have a known value for the period $T$ and the spring constant $k$, we can substitute these values into the rearranged formula to find the mass. For example, if $T = 1.5$ seconds and $k = 25 , \text{N/m}$, we substitute:

$$ m = \frac{(1.5)^2 \cdot 25}{4\pi^2} $$

4. Calculate the mass

Now we calculate the value using the provided values:

1. Calculate $T^2$:

$$ (1.5)^2 = 2.25 $$

2. Substitute back into the equation:

$$ m = \frac{2.25 \cdot 25}{4\pi^2} $$

3. Simplify and calculate the final mass.