How Does Cutting a Spring Affect SHM Frequency and Motion Characteristics?

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Cutting a uniform spring in half results in each half having a spring constant that is double that of the original spring, which affects the frequency of simple harmonic motion (SHM) by increasing it. The mass of the spring influences SHM characteristics, as it contributes to the total mass being accelerated, thereby affecting parameters like period and frequency. When considering a pendulum at different altitudes, gravity is weaker at mountain tops, which would cause the pendulum to take longer to complete a cycle, resulting in a gain of time compared to sea level. The discussion emphasizes the importance of understanding the relationship between spring constants, mass, and gravitational effects on motion. Overall, these concepts are crucial for analyzing SHM and pendulum behavior in varying conditions.
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Just going through some concept questions and these three have me stumped. If anyone can lend a hand.

  • If a uniform spring is cut in half, what is the force constant of each half? Justify your answer. How would the frequency of SHM using a half-spring difer from that using the same mass and the entire spring.
  • The analysis of SHM in this chapter ignored the mass of the spring. How does the spring's mass change the characteristics of motion?
  • Assuming a pendulum keeps perfect time at sea level, do you lose time, gain time or neither at a mountain top? (I want to say time runs slower from what I know of special relativity but what about just in classical physics?)
 
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godtripp said:
Just going through some concept questions and these three have me stumped. If anyone can lend a hand.

  • If a uniform spring is cut in half, what is the force constant of each half? Justify your answer. How would the frequency of SHM using a half-spring difer from that using the same mass and the entire spring.
  • The analysis of SHM in this chapter ignored the mass of the spring. How does the spring's mass change the characteristics of motion?
  • Assuming a pendulum keeps perfect time at sea level, do you lose time, gain time or neither at a mountain top? (I want to say time runs slower from what I know of special relativity but what about just in classical physics?)

For the first question, ask yourself these question: What does the spring constant tell you? (I'm assuming that by "force constant" you mean what some textbooks refer to as the "spring constant".) Is it an intrinsic property of the metal out of which the spring is made? Does the length of the spring enter into its definition?

For the second question, if you consider the mass of the spring in SHM does the total mass that is being accelerated going up or down? Where does mass enter into the parameters that characterize SHM, parameters like period, frequency, amplitude...

For the third question, What equation tells you the period of a pendulum? Two values enter into this expression. One is gravity, the other is the length of the pendulum. Is gravity stronger at sea level or on a mountain top? Why?
 
Thank you so much, this helped clear a lot of things up for me.
 
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