Exploring SHM and Spring Dynamics: Concept Questions and Answers

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In summary, the force constant of a half-spring is less than that of a whole spring due to the decreased mass of the half-spring. The frequency of SHM will be different depending on the mass of the spring, and the equation for the period of a pendulum is affected by both gravity and the length of the pendulum.
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godtripp
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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?
 
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Thank you so much, this helped clear a lot of things up for me.
 

What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion (SHM) is a type of periodic motion where an object oscillates back and forth between two points due to a restoring force that is directly proportional to its displacement from the equilibrium position. This motion is characterized by a sinusoidal pattern and is found in many natural systems, such as pendulums and springs.

What is the equation for SHM?

The equation for SHM is given by: x(t) = A*cos(ωt + φ), where x(t) is the displacement of the object from its equilibrium position at time t, A is the amplitude of the motion, ω is the angular frequency, and φ is the phase constant. This equation can also be expressed in terms of the object's velocity and acceleration.

How does the spring constant affect SHM?

The spring constant (k) is a measure of the stiffness of a spring and it directly affects the frequency of SHM. A higher spring constant results in a higher frequency of oscillation, while a lower spring constant results in a lower frequency. This means that stiffer springs will have a faster back and forth motion compared to more flexible springs.

What is the relationship between mass and SHM?

The mass of an object affects the period of SHM, which is the time it takes for one complete oscillation. The greater the mass, the longer the period will be. This is because a larger mass requires more force to move it, which results in a slower oscillation. However, the mass does not affect the frequency of SHM.

What factors can affect the amplitude of SHM?

The amplitude of SHM is affected by three main factors: the initial displacement of the object from its equilibrium position, the energy of the system, and the damping force. A larger initial displacement will result in a larger amplitude, while a smaller initial displacement will result in a smaller amplitude. The energy of the system also affects the amplitude, as a higher energy will result in a larger amplitude. Finally, a damping force, which is a resistive force that acts against the motion, can decrease the amplitude of SHM over time.

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