Question

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

Weight of remaining package ≈ 27.24 lb.

Step-by-step solution

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, $f=3\text{cycles/s}$: $\omega =2\pi (3)=6\pi \text{rad/s}$

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: $m=\frac{50}{(6\pi {)}^2}$

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: $W=(50/(6\pi {)}^2)\times 32.2$

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 = $\text{You can't use 'macro parameter character \#' in math mode}$ The remaining package weighs approximately #Weight# lb.

Key concepts

Angular Frequency

Angular frequency represents how fast something rotates or oscillates in a circle. For a spring-mass system, it helps us understand the vibration speed of the object. - The formula to calculate angular frequency is $\omega =2\pi f$, where $\omega$ is the angular frequency and $f$ is the frequency in cycles per second.- In our example, the system oscillates at 3 cycles per second, giving us $\omega =6\pi$ rad/s.This information is crucial for analyzing the system's behavior and determining other properties like mass.

Frequency Formula

The frequency formula for a spring-mass system connects the mass of the object and the stiffness of the spring. The formula is $f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$, where:

• $f$ is the frequency.
• $k$ is the stiffness of the spring.
• $m$ is the mass of the object.

To find the mass of the remaining package, the formula can be rearranged to solve for mass:$m=\frac{k}{(2\pi f{)}^2}$.In the example, using the given stiffness 50 lb/in and $\omega =6\pi$ rad/s, we can determine the mass of the remaining package in slugs.

Stiffness

Stiffness in a spring-mass system describes how resistant a spring is to being compressed or stretched. It is represented by the symbol $k$ and often measured in units of force per unit length, such as lb/in for our example. - The stiffness influences the system's oscillation frequency. Higher stiffness means a stronger spring, leading to higher frequency and faster oscillations.In our problem, the scale’s stiffness is 50 lb/in. This is an essential factor influencing how the system vibrates after one package is knocked off.

Weight Conversion

Weight conversion is required to obtain the actual weight in pounds from the mass obtained in slugs. Here are the steps:

• First, calculate the mass in slugs using the spring's stiffness and the angular frequency as derived earlier.
• Use the formula Weight = Mass $\times$ Gravity to convert mass to weight, where gravity is $32.2{\text{ft/s}}^2$.

For example, substituting the given values, we compute the weight before subtracting the scale's plate weight. The final step is to account for the plate's weight, ensuring that the correct weight of the remaining package is calculated.