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Ch 15: Oscillations
All textbooksKnight Calc 5th EditionCh 15: OscillationsProblem 15
Chapter 15, Problem 15

A compact car has a mass of 1200 kg. Assume that the car has one spring on each wheel, that the springs are identical, and that the mass is equally distributed over the four springs. b. What will be the car's oscillation frequency while carrying four 70 kg passengers?

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1
Calculate the total mass of the car including the passengers. Since there are four passengers each weighing 70 kg, the total additional mass is 4 times 70 kg. Add this to the original mass of the car.
Determine the effective mass supported by each spring. Since the car's mass is equally distributed over the four springs, divide the total mass of the car (including passengers) by four.
Use the formula for the frequency of oscillation for a mass-spring system, which is given by \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \), where \( f \) is the frequency, \( k \) is the spring constant, and \( m \) is the mass supported by each spring.
Since the spring constant \( k \) remains unchanged (as the springs themselves are not altered), use the effective mass calculated in step 2 for \( m \) in the formula.
Calculate the frequency using the values of \( k \) and \( m \) in the formula to find the oscillation frequency of the car with passengers.

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Key Concepts

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

Mass Distribution

In this scenario, the total mass of the car, including passengers, is distributed evenly across the four springs. Understanding how mass is distributed is crucial for calculating the effective mass that each spring supports, which directly influences the oscillation frequency of the system.
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Spring Constant and Oscillation Frequency

The oscillation frequency of a mass-spring system is determined by the spring constant (k) and the effective mass (m) supported by the springs. The formula for the frequency (f) is f = (1/2π)√(k/m). This relationship highlights how the stiffness of the springs and the mass they support affect the frequency of oscillation.
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Simple Harmonic Motion

The car's oscillation can be modeled as simple harmonic motion (SHM), where the restoring force is proportional to the displacement from equilibrium. In SHM, the system oscillates around a central position, and understanding this concept is essential for analyzing the behavior of the car as it moves over bumps or uneven surfaces.
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Related Practice
Textbook Question
A 200 g block hangs from a spring with spring constant 10 N/m. At t = 0 s the block is 20 cm below the equilibrium point and moving upward with a speed of 100 cm/s. What are the block's a. Oscillation frequency?
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Textbook Question
Interestingly, there have been several studies using cadavers to determine the moments of inertia of human body parts, information that is important in biomechanics. In one study, the center of mass of a 5.0 kg lower leg was found to be 18 cm from the knee. When the leg was allowed to pivot at the knee and swing freely as a pendulum, the oscillation frequency was 1.6 Hz. What was the moment of inertia of the lower leg about the knee joint?
383
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Textbook Question
A mass hanging from a spring oscillates with a period of 0.35 s. Suppose the mass and spring are swung in a horizontal circle, with the free end of the spring at the pivot. What rotation frequency, in rpm, will cause the spring's length to stretch by 15%?
685
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Textbook Question
It has recently become possible to 'weigh' DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P15.58 shows a thin rectangular cantilever etched out of silicon (density 2300 kg/m³) with a small gold dot (not visible) at the end. If pulled down and released, the end of the cantilever vibrates with SHM, moving up and down like a diving board after a jump. When bathed with DNA molecules whose ends have been modified to bind with gold, one or more molecules may attach to the gold dot. The addition of their mass causes a very slight—but measurable—decrease in the oscillation frequency. A vibrating cantilever of mass M can be modeled as a block of mass ⅓M attached to a spring. (The factor of ⅓ arises from the moment of inertia of a bar pivoted at one end.) Neither the mass nor the spring constant can be determined very accurately—perhaps to only two significant figures—but the oscillation frequency can be measured with very high precision simply by counting the oscillations. In one experiment, the cantilever was initially vibrating at exactly 12 MHz. Attachment of a DNA molecule caused the frequency to decrease by 50 Hz. What was the mass of the DNA?
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Textbook Question
A 100 g block attached to a spring with spring constant 2.5 N/m oscillates horizontally on a frictionless table. Its velocity is 20 c/m when 𝓍 = ─5.0 cm c. What is the block's position when the acceleration is maximum?
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Textbook Question
Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut then pulls her away from the wall and releases her. The spring's length as a function of time is shown in FIGURE P15.46. b. What is her speed when the spring's length is 1.2 m?

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