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Chapter 13: Problem 73

A \(2.00-\mathrm{kg}\) bucket containing 10.0 \(\mathrm{kg}\) of water is hanging from a vertical ideal spring of force constant 125 \(\mathrm{N} / \mathrm{m}\) and oscillating up and down with an amplitude of 3.00 \(\mathrm{cm} .\) Suddenly the bucket springs a leak in the bottom such that water drops out at a steady rate of 2.00 \(\mathrm{g} / \mathrm{s}\) . When the bucket is half full, find (a) the period of oscillation and (b) the rate at which the period is changing with respect to time. Is the period getting longer or shorter? (c) What is the shortest period this system can have?

Short Answer

Expert verified
The period decreases as water leaks; at half-full it's 1.49s. The shortest period is 0.80s.

Step by step solution

01

Define the System

The system consists of a bucket and water with a total mass hanging from a spring, oscillating with some amplitude. The bucket slowly loses water, affecting the mass and thus the period of oscillation.
02

Neglect Mass Change During Oscillation

Since the change in mass due to the leaking water is slow compared to the oscillation period, we assume constant mass for each period.
03

Determine Initial and Final Mass

The initial mass of the system is 12 kg (2 kg bucket + 10 kg water). When the bucket is half-full, the water mass is 5 kg. Thus, the final mass is 7 kg.
04

Formulate the Period of Oscillation

The period of oscillation for a mass-spring system is given by \( T = 2\pi \sqrt{\frac{m}{k}} \). Initially, \( m = 12 \) kg and \( k = 125 \, \mathrm{N/m} \). Calculate it for \( m = 7 \) kg.
05

Calculate Period for Full and Half-Full Bucket

For full bucket, \( T_{12} = 2\pi \sqrt{\frac{12}{125}} = 2\pi \sqrt{0.096} \approx 1.96 \, \mathrm{s} \). For half-full, \( T_{7} = 2\pi \sqrt{\frac{7}{125}} = 2\pi \sqrt{0.056} \approx 1.49 \, \mathrm{s} \).
06

Calculate Rate of Change of Period

Differentiate the period expression with respect to mass: \( \frac{dT}{dm} = \pi \sqrt{\frac{1}{mk}} \). Calculate when \( m \) is changing due to leakage.
07

Insert the Mass Loss Rate

Water leaks at 0.002 kg/s. For half-full, \( \frac{dT}{dt} = \frac{dT}{dm} \cdot \frac{dm}{dt} = \pi \sqrt{\frac{1}{125 \times 7}} \times (-0.002) \).
08

Evaluate the Change in Period

Calculate \( \frac{dT}{dt} \approx -0.00029 \, \mathrm{s}/\mathrm{s} \), indicating the period is decreasing.
09

Determine Shortest Period

The period is shortest as the bucket is completely empty. The smallest mass is 2 kg. Calculate \( T_{2} = 2\pi \sqrt{\frac{2}{125}} = 2\pi \sqrt{0.016} \approx 0.80 \, \mathrm{s} \).

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Key Concepts

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

Oscillation Period
The oscillation period is an essential concept in simple harmonic motion, especially for systems like a mass and spring. It represents the time it takes for one complete cycle of oscillation. For a mass-spring system, the period \( T \) is determined by the formula \( T = 2\pi \sqrt{\frac{m}{k}} \), where \( m \) is the mass of the object and \( k \) is the spring constant.

In the given problem, the initial mass of the system is 12 kg, consisting of a 2 kg bucket and 10 kg of water. As water leaks out, the mass decreases, impacting the oscillation period. Initially, the period is about 1.96 seconds for the full bucket. When the bucket is half-full, the mass is reduced to 7 kg, shortening the period to approximately 1.49 seconds.

This shows that the oscillation period is directly related to the mass; as mass decreases, the period becomes shorter.
Mass-Spring System
A mass-spring system is a classic model in physics used to study oscillations. It consists of a mass attached to a spring, which can stretch or compress. When displaced from its equilibrium position and released, the mass will oscillate around this point. This back-and-forth movement is characteristic of simple harmonic motion. The mass-spring system's behavior depends primarily on two factors: the mass of the object and the spring's force constant \( k \).

In our bucket-leaking scenario, the bucket and water serve as the mass. The spring has a force constant of 125 N/m. As the water leaks, the system's mass reduces, changing the period of the oscillations.

The bucket initially starts oscillating with a full load of water. As water leaks out at a steady rate, the changing mass affects the oscillation dynamics, illustrating how sensitive the system is to variations in mass.
Leakage Effect on Oscillation
When leakage is introduced to an oscillating mass-spring system, it affects the system's dynamics by gradually reducing the effective mass. The leakage rate in this scenario is steady at 2 g/s, equivalent to 0.002 kg/s.

While a single period's change might not be significant, continually losing mass will affect the system over time. This steady decrease in mass makes the oscillation period change, which can be analyzed by differentiating the period formula with respect to time. The rate of change is found to be approximately \(-0.00029 \text{ s/s}\).

This negative rate indicates that the period is decreasing as the bucket loses water, confirming that the system will oscillate faster as it becomes lighter. Eventually, when the bucket is completely empty, the period will be the shortest, calculated to be approximately 0.80 seconds. This illustrates how external factors like leakage can dynamically alter oscillation characteristics in a mass-spring system.

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Most popular questions from this chapter

A cheerleader waves her pom-pom in SHM with an amplitude of 18.0 \(\mathrm{cm}\) and a frequency of 0.850 \(\mathrm{Hz}\) . Find (a) the maximum magnitude of the acceleration and of the velocity; \((6)\) the acceleration and speed when the poin-poin's coordinate is \(x=+9.0 \mathrm{cm}\); (c) the time required to move from the equilibrium position directly to a point 12.0 \(\mathrm{cm}\) away. (d) Which of the quantities asproach used parts \((\mathrm{a}),(\mathrm{b}),\) and \((\mathrm{c})\) can be found using the energy approach used in Section \(13.3,\) and which cannot? Explain.

A 0.500 \(\mathrm{kg}\) mass on a spring has velocity as a function of time given by \(v_{x}(t)=(3.60 \mathrm{cm} / \mathrm{s}) \sin \left[\left(4.71 \mathrm{s}^{-1}\right) t-\pi / 2\right]\) . What are (a) the period; (b) the amplitude; (c) the maximum acceleration of the mass; (d) the force constant of the spring?

A large bell is hung from a wooden beam so it can swing back and forth with negligible friction. The center of mass of the bell is 0.60 \(\mathrm{m}\) below the pivot, the bell has mass 34.0 \(\mathrm{kg}\) , and the moment of inertia of the bell about an axis at the pivot is 18.0 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) The clapper is a small, 1.8-kg mass attached to one end of a slender rod that has length \(L\) and negligible mass. The other end of the rod is attached to the inside of the bell so it can swing freely about the same axis as the bell. What should be the length \(L\) of the clapper rod for the bell to ring silently - that is, for the period of oscillation for the bell to equal that for the clapper?

A certain alarm clock ticks four times each second, with each tick representing half a period. The balance wheel consists of a thin rim with radius \(0.55 \mathrm{cm},\) connected to the balance szaff by thin spokes of negligible mass. The total mass of the balance wheel is 0.90 \(\mathrm{g}\) . (a) What is the moment of inertia of the balance wheel about its shaft? (b) What is the torsion constant of the hairspring?

A rocket is accelerating upward at 4.00 \(\mathrm{m} / \mathrm{s}^{2}\) from the launchpad on the earth. Inside a small \(1.50-\mathrm{kg}\) ball hangs from the ceiling by a light \(1.10-\mathrm{m}\) wire. If the ball is displaced \(8.50^{\circ}\) from the vertical and released, find the amplitude and period of the resulting swings of this pendulum.
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