Ch. 14 - Oscillations

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 14 - Oscillations

Problem 67

Chapter 14, Problem 67

An 1150-kg automobile has springs with k = 14,000 N/m. One of the tires is not properly balanced; it has a little extra mass on one side compared to the other, causing the car to shake at certain speeds. If the tire radius is 42 cm, at what speed will the wheel shake most?

Verified step by step guidance

Step 1: Recognize that the problem involves a resonance phenomenon. The car will shake most at the natural frequency of the spring-mass system. The natural frequency can be calculated using the formula $ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} $, where $ k $ is the spring constant and $ m $ is the mass of the automobile.

Step 2: Convert the given spring constant $ k = 14,000 \, \text{N/m} $ and mass $ m = 1150 \, \text{kg} $ into the formula. Substitute these values into $ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} $ to find the natural frequency of the system.

Step 3: Once the natural frequency $ f $ is determined, calculate the angular frequency $ \omega $ using the relationship $ \omega = 2\pi f $. This angular frequency corresponds to the rate at which the system oscillates.

Step 4: Relate the angular frequency $ \omega $ to the speed of the wheel. The speed $ v $ at which the wheel shakes most can be calculated using $ v = \omega \cdot r $, where $ r $ is the radius of the tire. Convert the radius $ r = 42 \, \text{cm} $ into meters ($ r = 0.42 \, \text{m} $) before substituting into the formula.

Step 5: Substitute the calculated angular frequency $ \omega $ and the radius $ r $ into $ v = \omega \cdot r $ to determine the speed at which the wheel shakes most. Ensure all units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes the oscillatory motion of an object where the restoring force is directly proportional to the displacement from its equilibrium position. In this context, the shaking of the automobile due to the unbalanced tire can be modeled as SHM, where the frequency of oscillation is influenced by the mass distribution and spring constant.

Natural Frequency

Natural frequency is the frequency at which a system tends to oscillate in the absence of any driving force. For the automobile's suspension system, the natural frequency can be calculated using the mass of the car and the spring constant. When the car reaches a speed that matches this frequency, resonance occurs, leading to increased shaking.

Resonance

Resonance is a phenomenon that occurs when a system is driven at its natural frequency, resulting in large amplitude oscillations. In the case of the automobile, if the speed of the car matches the natural frequency of the tire's oscillation due to the imbalance, the shaking will be most pronounced, potentially causing discomfort and instability.

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