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Chapter 14: Problem 1

For the damped pendulum, a. Assume that \(F\) is proportional to \(v^{2}\) and use dimensional analysis to show that \(t=\) \(\sqrt{r / g} h\left(\theta, r k_{1} / m\right)\) b. Assume that \(F\) is proportional to \(v^{2}\) and describe an experiment to test the model \(t=\sqrt{r / g} h\left(\theta, r k_{1} / m\right)\)

Short Answer

Expert verified
Use dimensional analysis to express time as \( t=\sqrt{r/g} h(\theta, rk_1/m) \), and test using varied pendulum setups to compare time decay data with model predictions.

Step by step solution

01

Define the Problem

For a damped pendulum with damping force proportional to the square of velocity \( F \propto v^2 \), use dimensional analysis to determine the relationship between time \( t \), radius \( r \), gravitational acceleration \( g \), angle \( \theta \), damping constant \( k_1 \), and mass \( m \).
02

Identify Critical Variables

Consider the variables involved: time \( t \), radius \( r \), gravitational acceleration \( g \), damping constant \( k_1 \), mass \( m \), and angle \( \theta \). The units of these variables are: \([t] = T\), \([r] = L\), \([g] = L/T^2\), \([k_1] = M/T\), \([m] = M\), and angle \( \theta \) is dimensionless.
03

Express Dimensional Relationships

Construct dimensionless groups by equating the dimensional parts on both sides:\[ [t] = \sqrt{\frac{[r]}{[g]}} = \left(\frac{L}{L/T^2}\right)^{1/2} = T \]\( h \left( \theta, \frac{r k_1}{m} \right) \) must contain combinations of \( \theta \) and \( \frac{rk_1}{m} \) such that the entire expression remains dimensionless.
04

Correlate Variables

To align with dimensional consistency, identify that \( t = \sqrt{r/g} \cdot h(\theta, rk_1/m) \) matches dimensions with the prior step:- \( h \left( \theta, \frac{rk_1}{m} \right) \) is a non-dimensional function due to \( rk_1/m = (L \cdot MT^{-1})/M = LT^{-1}. \) Since \( h \) itself must be non-dimensional, it transforms the problem into terms suitable for experimental validation.
05

Design the Experiment

For the experimental model test, use a physical pendulum setup:1. Vary \( r \), the pendulum arm length, and observe time period \( t \) changes.2. Measure \( \theta \), initial angular displacement.3. Adjust damping level via \( k_1 \), the dampening medium.4. Record pendulum decay times for different \( rk_1/m \) ratios.5. Compare experimental \( t \) values against theoretical predictions \( \sqrt{r/g} \cdot h(\theta, rk_1/m) \).
06

Analyze Experimental Data

Determine if experimental results align with the hypothesized model through comparison of empirical data points and theoretical functions for \( t \). Adjust model parameters as necessary based on empirical observations.

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Key Concepts

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

Damped Pendulum
A damped pendulum is a swinging motion device that experiences a force trying to slow it down. Imagine a playground swing that doesn't keep going forever because of a breeze stopping it. That's damping in action! The damping force in our scenario is proportional to the square of the velocity. Meaning, as the pendulum swings faster, the force trying to stop it gets even stronger. This force impacts the time period鈥攁 measure of how long it takes for the pendulum to complete one swing. Understanding the time period is important because it tells us how the pendulum behaves under different conditions, like changing the length of its arm or how much air is trying to slow it down.
Model Testing
Model testing is about seeing if our ideas really work. In this case, it means checking if our formula for the damped pendulum makes sense in the real world. We can do this by creating a physical pendulum and changing some of its features, like the length of the arm or how many weights are pulling it down. For example, if we think that changing these things should make the pendulum take longer or shorter to swing, we can test that by timing how long it swings. If our formula predicts the times correctly, then we know our model is good. However, if the times are way off, then maybe our formula needs to be tweaked.
Damping Force
Damping force is like a speed-breaker for a pendulum. It's a force that slows down motion, ensuring the pendulum doesn't swing forever. In this case, it's related to how fast the pendulum is moving鈥攕pecifically, its velocity squared. When the pendulum moves quickly, the damping force is strong, acting like an invisible hand gently pushing against it. This force is not a mystery; it happens because of air resistance or friction. It's essential to consider damping force when analyzing how a pendulum moves because it influences the speed and distance of each swing. By recognizing this force, we can predict how the pendulum will stop or slow over time.
Experimental Design
Designing an experiment is like creating a recipe to see if our scientific hunch is correct. For the damped pendulum, this involves setting up a scenario where we can easily change a few variables and measure how they affect the pendulum's motion. Here's what we might do:
  • Change the pendulum's arm length and note how the swinging time varies.
  • Adjust the initial angle from which the pendulum is released.
  • Control the damping level by adding different materials to slow it down.
  • Calculate and compare the time it takes for the pendulum to slow under various settings.
  • Finally, compare these real-life measurements with what our math formula predicted.
By doing these steps, we gather valuable data, testing if our hypothesis鈥攐r guess鈥攊s on point, or if we need to think of other factors influencing the outcome.

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Most popular questions from this chapter

Using dimensional analysis, find a proportionality relationship for the centrifugal force \(F\) of a particle in terms of its mass \(m\), velocity \(v\), and radius \(r\) of the curvature of its path.

Two smooth balls of equal weight but different diameters are dropped from an airplane. The ratio of their diameters is \(5 .\) Neglecting compressibility (assume constant Mach number), what is the ratio of the terminal velocities of the balls? Are the flows completely similar?

Determine whether the equation $$ s=s_{0}+v_{0} t-0.5 g t^{2} $$ is dimensionally compatible, if \(s\) is the position (measured vertically from a fixed reference point) of a body at time \(t, s_{0}\) is the position at \(t=0, v_{0}\) is the initial velocity, and \(g\) is the acceleration caused by gravity.

a. Use dimensional analysis to establish the cube-root law $$ r\left(\frac{\rho}{W}\right)^{1 / 3}=\text { constant } $$ for scaling of explosions, where \(r\) is the radius or depth of the crater, \(\rho\) is the density of the soil medium, and \(W\) the mass of the explosive. b. Use dimensional analysis to establish the one-fourth-root law $$ r\left(\frac{\rho g}{E}\right)^{1 / 4}=\text { constant } $$ for scaling of explosions, where \(r\) is the radius or depth of the crater, \(\rho\) is the density of the soil medium, \(g\) is gravity, and \(E\) is the charge energy of the explosive.

The height \(h\) that fluid will rise in a capillary tube decreases as the diameter \(D\) of the tube increases. Use dimensional analysis to determine how \(h\) varies with \(D\) and the specific weight \(w\) and surface tension \(\sigma\) of the liquid.
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