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Chapter 14: Problem 68

Four passengers with combined mass 250 kg compress the springs of a car with worn-out shock absorbers by 4.00 \(\mathrm{cm}\) when they get in. Model the car and passengers as a single body on a single ideal spring. If the loaded car has a period of vibration of 1.92 s, what is the period of vibration of the empty car?

Short Answer

Expert verified
The period of vibration for the empty car is shorter than 1.92 s.

Step by step solution

01

Understand the Problem

We have a car and passengers modeled as a single mass on a spring. The car compresses the spring when passengers get in, and we're given the period of vibration for this loaded car, which includes the passengers' mass. We need to determine the period of vibration of the car when empty, removing the passengers' mass.
02

Identify Known Variables

The mass of the passengers is 250 kg, the displacement (compression) is 4.00 cm (0.04 m), and the period of vibration of the loaded car is 1.92 s. We need to find the period of vibration for just the car.
03

Calculate the Spring Constant

Using Hooke's Law, calculate the spring constant \( k \). The force exerted by the passengers is \( F = mg \), where \( m = 250 \) kg and \( g = 9.8 \) m/s虏. Solve for \( k \) using the equation \( F = kx \), where \( x = 0.04 \) m. This results in \( k = \frac{mg}{x} = \frac{250 \times 9.8}{0.04} \) N/m.
04

Determine Total Mass of Loaded Car

The loaded car's period \( T = 1.92 \) s gives us an expression involving the total mass \( m + M \) (where \( M \) is the mass of the car) using the formula for the period \( T = 2 \pi \sqrt{\frac{m+M}{k}} \). Rearrange to solve for \( m + M \).
05

Rearrange for Car's Mass

From Step 3, solve \( m + M = \frac{kT^{2}}{4\pi^{2}} \). With the previously calculated \( k \) and \( T = 1.92 \), find \( m + M \). Then subtract the known mass of passengers \( m = 250 \) kg to find the mass \( M \) of the car.
06

Calculate Period of Empty Car

With the mass of the car \( M \) known, find the period of the empty car using \( T_{empty} = 2 \pi \sqrt{\frac{M}{k}} \). Substitute \( M \) and \( k \) to find the new period.

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Key Concepts

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

Vibration Period
The vibration period of a system, such as a loaded car modeled as a mass on a spring, describes how long it takes for one complete oscillation of the system. When the car is loaded, this period is 1.92 seconds. It means that with the added mass of passengers, the time for oscillation is longer due to increased inertia. To calculate the vibration period of an empty car, follow:
  • Determine the spring constant using external forces.
  • Find the effective mass during the loaded period using the period formula.
  • Use the effective mass of the car alone to find the period of the empty car using the same period formula.
Remember, a stiffer spring or lighter mass decreases the vibration period, making oscillations quicker.
Hooke's Law
Hooke鈥檚 Law is fundamental when dealing with springs and describes the relationship between force and spring displacement. The law is expressed as \( F = kx \), where:
  • \( F \) is the force applied to the spring.
  • \( k \) is the spring constant, indicating spring stiffness.
  • \( x \) is the displacement caused by the force.
In our exercise, Hooke's Law is used to find the spring constant. The force exerted by the passengers is calculated by multiplying their mass by gravitational acceleration: \( F = 250 \times 9.8 \) N. This force is then used to determine the spring constant \( k \) using displacement \( x = 0.04 \) m. Calculating \( k \) provides insight into how the spring responds to the mass and implies the potential for vibration.
Car Mass Calculation
Finding the car's mass involves a few essential steps. First, use the known vibration period of the loaded car (1.92 seconds) and the spring constant, calculated using Hooke's Law, in the formula for the period of a mass-spring system:\[T = 2 \pi \sqrt{\frac{m + M}{k}}\]Here, \( m \) is the passenger mass, and \( M \) is the car's mass. By rearranging this formula, one can solve for the total mass \( m + M \). Subtract the passenger mass from this sum to isolate \( M \). Finally, with the car's mass alone, the period of the empty car is recalculated. Repeat the process using its mass and known spring constant:\[T_{empty} = 2 \pi \sqrt{\frac{M}{k}}\]This altered period will typically be less since the car is now lighter, showcasing the dynamic interplay between mass and vibration in mechanical systems.

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Most popular questions from this chapter

You want to find the moment of inertia of a complicated machine part about an axis through its center of mass. You suspend it from a wire along this axis. The wire has a torsion constant of 0.450 \(\mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\) . You twist the part a small amount about this axis and let it go, timing 125 oscillations in 265 s. What is the moment of inertia you want to find?

BID (a) Music. When a person sings, his or her vocal cords vibrate in a repetitive pattern that has the same frequency as the note that is sung. If someone sings the note \(B\) flat, which has a fre- quency of 466 Hz, how much time does it take the person's vocal cords to vibrate through one complete cycle, and what is the angu- lar frequency of the cords? (b) Hearing. When sound waves strike the eardrum, this membrane vibrates with the same frequency as the sound. The highest pitch that typical humans can hear has a period of 50.0\(\mu \mathrm{s} .\) What are the frequency and angular frequency of the vibrating eardrum for this sound? (c) Vision. When light having vibrations with angular frequency ranging from \(2.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) to \(4.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light? (d) Ultrasound. High-frequency sound waves (ultrasound) are used to probe the interior of the body, much as x rays do. To detect small objects such as tumors, a frequency of around 5.0 \(\mathrm{MHz}\) is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?

A 0.150 -kg toy is undergoing SHM on the end of a horizontal spring with force constant \(k=300 \mathrm{N} / \mathrm{m} .\) When the object is 0.0120 \(\mathrm{m}\) from its equilibrium position, it is observed to have a speed of 0.300 \(\mathrm{m} / \mathrm{s} .\) What are (a) the total energy of the object at any point of its motion; (b) the amplitude of the motion; \((\mathrm{c})\) the maximum speed attained by the object during its motion?

Don't Miss the Boat. While on a visit to Minnesota ("Land of \(10,000\) Lakes"), you sign up to take an excursion around one of the larger lakes. When you go to the dock where the \(1500-\mathrm{kg}\) boat is tied, you find that the boat is bobbing up and down in the waves, executing simple harmonic motion with amplitude 20 \(\mathrm{cm} .\) The boat takes 3.5 \(\mathrm{s}\) s to make one complete up-and-down cycle. When the boat is at its highest point, its deck is at the same height as the stationary dock. As you watch the boat bob up and down, you (mass 60 \(\mathrm{kg}\) ) begin to feel a bit woozy, due in part to the previous night's dinner of lutefisk. As a result, you refuse to board the boat unless the level of the boat's deck is within 10 \(\mathrm{cm}\) of the dock level. How much time do you have to board the boat comfortably during each cycle of up-and-down motion?

A Pendulum on Mars. A certain simple pendulum has a period on the earth of 1.60 s. What is period on the surface of Mars, where \(g=3.71 \mathrm{m} / \mathrm{s}^{2} ?\)
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