How Do You Calculate Mass and Acceleration in a Spring Mass System?

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To calculate the mass of an object in a spring-mass system with a spring constant of 2.56 N/m and a frequency of 0.148 Hz, the period T is first determined as T = 1/f, resulting in T = 6.76 seconds. Using the formula m = Tk/4π², the mass is calculated to be approximately 2.96 kg. For acceleration at a displacement of -0.120 m, the equation a = -kx/m is applicable, where k is the spring constant and x is the displacement. The discussion highlights confusion regarding the correct application of formulas for both mass and acceleration. Ultimately, the calculations confirm the correct approach to solving the problem.
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Homework Statement



An object suspended from a spring with a spring constant of 2.56 N/m vibrates with a frequency of 0.148 Hz
a. What is the mass of the object
b. What is the acceleration of the object at a displacement of -0.120m from the equilibriam position

Homework Equations


a. T=2pi\sqrt{}m/k m=Tk/4pi2' T=1/f
b. a=-kx/m ?

The Attempt at a Solution


a.T=1/f T=1/0.148Hz T=6.756756757s m=Tk/4pi2' m=(6.756756757s)(2.56N/m)/4pi2' m=2.96kg ---- I do not understand if this is right and I think I made a mistake ----
b. I can't figure out what equation to use and how to manipulate it to get an equation for acceleration for the condition
 
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a. It's right.
b. It's a=-kx/m.
 
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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