Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 17

A steel ball of mass \(10 \mathrm{~g}\) is placed in the tube of cross-sectional area \(1 \mathrm{~cm}^{2}\) in Rüchhardt's apparatus. The tube is connected to a jar of air having a capacity of 5 liters, the pressure of the air being \(76 \mathrm{~cm} \mathrm{Hg}\). (a) What is the period of vibration for the ball? (b) If the ball is held initially at a position where the air pressure is exactly atmospheric and then allowed to fall, how far will the ball drop before it starts to come up?

Short Answer

Expert verified
The period of vibration for the ball is given by T = 2π√(m/k).The ball will drop a distance x = mg/k before starting to come up

Step by step solution

01

- Convert Units

First, convert all given units to the SI system. Mass of the ball: 1.0 g = 0.01 kg. Cross-sectional area of the tube: 1 cm² = 0.0001 m². Capacity of the jar: 5 liters = 0.005 m³. Pressure of air: 76 cm Hg = 76 × 133.322 = 10132.872 Pa.
02

- Use Ideal Gas Law

Calculate the number of moles of air in the jar using the ideal gas law: PV = nRT, where:P = 10132.872 PaV = 0.005 m³R = 8.314 J/(mol·K)T (assuming room temperature) = 298 K. Thus, n = PV / RT.
03

- Determine the Air Stiffness

Determine the stiffness, k, of the air column. The formula is: k = (P×A²)/V, where: A = 0.0001 m² P = 10132.872 Pa V = 0.005 m³. Calculate k using the given values.
04

- Calculate the Period of Vibration

Using the stiffness (k) from the previous step, calculate the period of vibration (T) of the ball using the formula: T = 2π√(m/k), where m = 0.01 kg and k is the value got from Step 3.
05

- Determine the Initial Amplitude

To find how far the ball will drop before starting to come up, use Hooke's Law: F = kx, where F is the force due to gravity (mg), k is the stiffness calculated in Step 3, and x is the displacement (drop distance). Solve for x: mg = kx x = mg/k.
06

- Calculate the Displacement

Substitute the known values into the equation from Step 5 to determine the drop distance x. Use m = 0.01 kg, g = 9.81 m/s², and k from Step 3

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Ideal Gas Law
The ideal gas law is a fundamental equation in physics and chemistry, expressed as \( PV = nRT \). It relates the pressure (P), volume (V), and temperature (T) of an ideal gas with its number of moles (n) and the gas constant (R).
In our exercise, we are given the pressure (76 cm Hg), volume (5 liters), and assuming a room temperature of 298 K.
To use the ideal gas law, we first convert these values to SI units:
  • Pressure: 76 cm Hg = 76 × 133.322 = 10132.872 Pa
  • Volume: 5 liters = 0.005 m³
Using these values in the ideal gas law, we can solve for the number of moles (n) of air in the jar: \( n = \frac{PV}{RT} \). This calculation is vital for determining the stiffness of the air column in the subsequent steps.
Simple Harmonic Motion
Simple harmonic motion (SHM) describes a type of periodic motion where an object oscillates back and forth over the same path. This motion is characterized by its frequency and period.
In our exercise, the steel ball in Rüchhardt's apparatus exhibits SHM as it moves up and down due to the restoring force from the air compression and the gravitational pull. The key parameter to calculate here is the period (T) of the oscillation, which will give us how long it takes for one complete cycle of motion.
Hooke's Law
Hooke's Law states that the force needed to extend or compress a spring by some distance (x) is proportional to that distance. It is given by \( F = kx \), where F is the force, k is the stiffness constant, and x is the displacement.
In the context of Rüchhardt's apparatus, the stiffness constant (k) of the air column can be determined using the formula: \( k = \frac{PA^{2}}{V} \). Once we have the value of k, we can use Hooke's Law to calculate how far the ball will drop before it starts to come up: \( mg = kx \). Solving for x will give us the initial displacement caused by gravity.
Period of Vibration
The period of vibration (T) is the time it takes for one complete cycle of oscillation. It can be calculated using the formula \( T = 2\pi \sqrt{\frac{m}{k}} \), where m is the mass of the oscillating object and k is the stiffness constant.
In our problem, after determining k from previous steps, we substitute the mass of the ball (0.01 kg) and the calculated stiffness (k) into the equation to find the period of vibration. This will tell us precisely the duration of each oscillation cycle of the steel ball in the air column.
Unit Conversion
Unit conversion is crucial when dealing with physics problems to ensure consistency and accuracy.
In the given exercise, we performed various unit conversions:
  • Mass of the ball: 10 g = 0.01 kg
  • Cross-sectional area of the tube: 1 cm² = 0.0001 m²
  • Capacity of the jar: 5 liters = 0.005 m³
  • Pressure of air: 76 cm Hg = 10132.872 Pa
These conversions allow us to use the SI units required for the mathematical formulas and ensure that all other calculations are correct and meaningful. This step is often one of the first and most critical parts of solving such problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A thick-walled insulated chamber contains \(n_{i}\) moles of helium at high pressure \(P_{i}\). It is connected through a valve with a large, almost empty container of helium at constant pressure \(P_{0}\), very nearly atmospheric. The valve is opened slightly, and the helium flows slowly and adiabatically into the container until the pressures on the two sides of the valve are equal. Assuming the helium to behave like an ideal gas with constant heat capacities, show that: (a) The final temperature of the gas in the chamber is $$ T_{f}=T_{i}\left(\frac{P_{f}}{P_{i}}\right)^{(\gamma-1) / \gamma} $$ (b) The number of moles left in the chamber is $$ n_{f}=n_{i}\left(\frac{P_{f}}{P_{i}}\right)^{1 / \gamma} \text { . } $$ (c) The final temperature of the gas in the container is $$ T_{f}=\frac{T_{i}}{\gamma} \frac{1-P_{f} / P_{i}}{1-\left(P_{f} / P_{i}\right)^{1 / \gamma}} . $$

(a) Derive the following formula for a quasi-static adiabatic process for the ideal gas, assuming \(\gamma\) to be constant: $$ \frac{T}{P(\gamma-1) / \gamma}=\text { const. } $$ (b) Helium \(\left(\gamma=\frac{5}{3}\right)\) at \(300 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure is compressed quasi-statically and adiabatically to a pressure of 5 atm. Assuming that the helium behaves like the ideal gas, calculate the final temperature.

A cylindrical cocktail glass \(15 \mathrm{~cm}\) high and \(35 \mathrm{~cm}^{2}\) in cross section contains water up to the \(10-\mathrm{cm}\) mark. A card is placed over the top and held there while the glass is inverted. When the support for the card is removed, what mass of water must leave the glass in order that the rest of the water will remain in the glass, if one neglects the weight of the card? (Caution: Try this over a sink.)

The temperature of an ideal gas in a tube of very small, constant cross- sectional area varies linearly from one end \((x=0)\) to the other end \((x=L)\) according to the equation $$ T=T_{0}+\frac{T_{L}-T_{0}}{L} x $$ If the volume of the tube is \(V\) and the pressure \(P\) is uniform throughout the tube, show that the equation of state for \(n\) moles of gas is given by $$ P V=n R \frac{T_{L}-T_{0}}{\ln \left(T_{L} / T_{0}\right)} $$ Show that, when \(T_{L}=T_{0}=T\), the equation of state reduces to the obvious one, \(P V=n R T\)

(a) Show that the heat transferred during an infinitesimal quasi-static process of an ideal gas can be written $$ \mathrm{d} Q=\frac{C_{V}}{n R} V d P+\frac{C_{P}}{n R} P d V $$ Applying this equation to an adiabatic process, show that \(P V^{\prime}=\) const. (b) An ideal gas of volume \(0.05 \mathrm{ft}^{3}\) and pressure \(120 \mathrm{lb} / \mathrm{in}^{2}\) undergoes a quasi-static adiabatic expansion until the pressure drops to \(15 \mathrm{lb} / \mathrm{in}^{2}\). Assuming \(\gamma\) to remain constant at the value \(1.4\), calculate the final volume. Calculate the work.
See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Radiation

Read Explanation

Famous Physicists

Read Explanation

Modern Physics

Read Explanation

Work Energy and Power

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free