Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Chapter 14, Problem 9P
To determine
The frequency of the simple pendulum when the mass on the spring is added.
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A mass of 1kg is attached to a spring with a spring constant of k=10 N/m. you stretch the spring to distance A from equilibrium and let go. The mass oscillates with a certain period, frequency, and total energy. Now stretch the spring to a distance of 3A. compared to the original total energy E, what is the new total energy of the system now?
We will now determine how the 1/3 rule comes about.
Consider a spring of mass ms which is attached to a wall and oscillates on a frictionless surface as shown below. The spring’s mass is uniformly distributed along the length of the spring.
We will start with the infinitesimal form of kinetic energy, i.e. dKE = ½ (dms )v2. This formula will apply to an infinitesimal segment of the spring of length dx and mass dms as indicated below.
For any point on the spring, the velocity of oscillation will be given by v = (ve/L)x where ve is the velocity of the spring at its end where the mass m is attached, and L is the stretched length of the spring at that instant. Thus, when x = 0 then v = 0, and when x = L/2 then v = ½ ve.
Hint: Figure out how to relate dms to dx and then integrate both sides of the infinitesimal kinetic energy equation to get an equation for the kinetic energy of the spring that includes ms/3.
One astronaut on the Moon has a spring with a known spring constant k while another astronaut has simply a string of known length l. An interesting rock is found. Describe how the astronaut with spring can determine the mass of the rock from observations and data secured with the spring without knowing the magnitude of the acceleration due to gravity on the Moon. Is it possible for the astronaut with the string to check the result? Explain how or why not.
Chapter 14 Solutions
Physics for Scientists and Engineers
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