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Chapter 14: Problem 17

Mass-spring systems and pendulum systems can both be used in mechanical timing devices. What are the advantages of using one type of system rather than the othes in a device designed to generate reproducible time measurements over an extended period of time?

Short Answer

Expert verified
Answer: The advantages of using a mass-spring system include simpler design, easier miniaturization, and suitability for small devices. Pendulum systems can provide higher precision and be less sensitive to certain external factors with careful design but are typically suited for larger devices. The choice between the two depends on the specific requirements of the device.

Step by step solution

01

Understand mass-spring systems

Mass-spring systems consist of a mass (m) attached to a spring with spring constant k. When the mass is displaced from its equilibrium position and released, it oscillates back and forth around the equilibrium point with a time period (T) given by T = 2π√(m/k). This time period depends on the mass and the spring constant.
02

Understand pendulum systems

Pendulum systems consist of a weight suspended from a pivot, such that it can swing back and forth. The time period of a simple pendulum (a mass attached to a string of length L) is given by T = 2π√(L/g), where g is the acceleration due to gravity. The time period depends on the length of the pendulum and the local acceleration due to gravity.
03

Precision and external influences

Both mass-spring and pendulum systems are vulnerable to external influences, but in different ways. Mass-spring systems can be sensitive to changes in temperature, which affect the spring constant and thus the time period. Pendulum systems are affected by changes in the local acceleration due to gravity (e.g., due to altitude or latitude) and air resistance. However, a well-designed pendulum system can be less sensitive to these factors by using a temperature-insensitive material and operating in a vacuum.
04

Design Complexity

Mass-spring systems can be relatively easy to miniaturize and assemble, making them suitable for small devices like wristwatches and clocks. On the other hand, pendulum systems often require a larger structure, making them more suitable for larger devices like grandfather clocks.
05

Conclusion

The advantages of using a mass-spring system in a timing device include simpler design and the possibility to miniaturize the system. Pendulum systems can provide higher precision and insensitivity to certain external factors if designed carefully. The choice between a mass-spring or a pendulum system depends on the specific requirements of the device, such as size constraints and the desired level of precision.

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Most popular questions from this chapter

What is the period of a simple pendulum that is \(1.00 \mathrm{~m}\) long in each situation? a) in the physics lab b) in an clevator accelerating at \(2.10 \mathrm{~m} / \mathrm{s}^{2}\) upward c) in an elevator accelerating \(2.10 \mathrm{~m} / \mathrm{s}^{2}\) downward d) in an elevator that is in free fall

A \(3.00-\mathrm{kg}\) mass attached to a spring with \(k=140 . \mathrm{N} / \mathrm{m}\) is oscillating in a vat of oil, which damps the oscillations. a) If the damping constant of the oil is \(b=10.0 \mathrm{~kg} / \mathrm{s}\), how long will it take the amplitude of the oscillations to decrease to \(1.00 \%\) of its original value? b) What should the damping constant be to reduce the amplitude of the oscillations by \(99.0 \%\) in 1.00 s?

14.10 A pendulum is suspended from the ceiling of an elevator. When the elevator is at rest, the period of the pendulum is \(T\). The elevator accelerates upward, and the period of the pendulum is then a) still T. b) less than \(T_{-}\) c) greater than \(T\).

The figure shows a mass \(m_{2}=20.0\) g resting on top of a mass \(m_{1}=20.0 \mathrm{~g}\) which is attached to a spring with \(k=10.0 \mathrm{~N} / \mathrm{m}\) The coefficient of static friction between the two masses is 0.600 . The masses are oscillating with simple harmonic motion on a frictionless surface. What is the maximum amplitude the oscillation can have without \(m_{2}\) slipping off \(m_{1} ?\)

The period of a pendulum is \(0.24 \mathrm{~s}\) on Earth. The period of the same pendulum is found to be 0.48 s on planet \(X,\) whose mass is equal to that of Earth. (a) Calculate the gravitational acceleration at the surface of planet \(X\). (b) Find the radius of planet \(\mathrm{X}\) in terms of that of Earth.
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