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Two packages are placed on a spring scale whose plate weighs 10lb and whose stiffness is 50lb/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
Weight of remaining package ≈ 27.24 lb.

Step by step solution

01

Convert the frequency to angular frequency

To find the angular frequency, we can use the following formula: ω=2πf Substitute the given frequency value, f=3cycles/s: ω=2π(3)=6πrad/s
02

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=12πkm After rearranging, we get m=k(2πf)2 Using the given stiffness (lb/in) k=50 and the angular frequency (rad/s) ω=6π, we can find the mass (in slugs) as follows: m=50(6π)2
03

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 × Gravity, where gravity is 32.2ft/s2, and 1 slug is equivalent to 32.174 lb. Plug in the mass, m, and the gravity constant, g=32.2ft/s2, to get the weight: W=(50/(6π)2)×32.2
04

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/(6π)2)×32.2)10 Evaluate the expression and round your answer: Weight of remaining package = You can't use 'macro parameter character #' in math mode The remaining package weighs approximately #Weight# lb.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 ω=2πf, where ω 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 ω=6π 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=12πkm, 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=k(2πf)2.In the example, using the given stiffness 50 lb/in and ω=6π 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 × Gravity to convert mass to weight, where gravity is 32.2ft/s2.
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.

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Most popular questions from this chapter

A 12 kg spool that is 1 m in radius is pinned to a viscoelastic rod of negligible mass with effective properties k=10 N/m and c=8 Nsec/m. The end of the rod is attached to a rigid support as shown. Determine the natural frequency of the system if the spool rolls without slipping.

The cranking device shown consists of a mass-spring system of stiffiness 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 IO, determine the natural frequency of the system. (The spring is unstretched when connecting pin is directly over hub ' O '.)

A single degree of freedom system is represented as a 4 kg mass attached to a spring possessing a stiffness of 6 N/m and a viscous damper whose coefficient is 1 Nsec/m. (a) Determine the response of the horizontally configured system if the mass is displaced 2 meters to the right and released with a velocity of 4 m/sec. Plot and label the response history of the system. (b) Determine the response and plot its history if the damping coefficient is 5 Nsec/m. (c) Determine the response and plot its history if the damping coefficient is 10 Nsec/m.

Determine the overshoot of the system of Problem 2.25 if it is critically damped and v0=4 m/sec.

A single degree of freedom system is represented as a 4 kg mass attached to a spring possessing a stiffness of 6 N/m. If the coefficients of static and kinetic friction between the mass and the surface it moves on are μs=μk=0.1, and the mass is displaced 2 meters to the right and released with a velocity of 4 m/sec, determine the time after release at which the mass sticks and the corresponding displacement of the mass.

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