Solving for Spring Constant: Displacement and Time Relationship

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SUMMARY

The discussion focuses on calculating the spring constant (k) for a mass-spring system using the displacement and time relationship. Given a mass of 14.3 g and a frequency of 1.45 cycles/sec, the spring constant is determined to be 1.19 N/m using the formula k = m * (2 * pi * f)^2. Additionally, the general equation of oscillation is provided as x = x0*sin(ωt + φ), where the phase constant φ can be calculated using the initial conditions.

PREREQUISITES
  • Understanding of harmonic motion and oscillation principles
  • Familiarity with the spring constant formula k = m * (2 * pi * f)^2
  • Knowledge of trigonometric functions in oscillatory motion
  • Basic skills in graph interpretation related to displacement and time
NEXT STEPS
  • Study the derivation of the spring constant from Hooke's Law
  • Learn about the relationship between frequency and period in oscillatory systems
  • Explore the concept of phase constant in harmonic motion
  • Investigate the effects of mass and spring constant on oscillation frequency
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and oscillatory motion, as well as educators seeking to explain the principles of spring dynamics.

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Homework Statement



The graph of displacement versus time for a small mass at the end of a spring is shown. At t = 0, x = 0.43 cm. (a) If m = 14.3 g, find the spring constant, k. (b) Write the equation for displacement x as a function of time.

Homework Equations



f = (1/2pi) * SQRT (k/m)
f = 1/Period = 1/ 0.69 sec = 1.45 cycles/sec
SQRT (k/m) = 2 * pi * f

k = m * (2 * pi * f )^2

k = 0.0143 kg * (2 * pi * 1.45 cycles/sec
)^2


The Attempt at a Solution



1.19 N/m
 
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graph : http://www.webassign.net/userimages/mowat@ncsu/FG14_029.JPG

SORRY!
 
Your k value is correct.
The general equation of oscillation is
x = x0*sin(ωt + φ)
When t = 0, x = x0*sin(φ)
Put the values of the x and xo and find the value of φ. Then write the general equation.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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