Calculating Spring Constant for a Mass-Spring System: Technician's Experiment

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The discussion focuses on calculating the spring constant for a mass-spring system involving a bullet and a wooden block. A technician named Michelle conducts an experiment where a bullet strikes a block attached to a spring, compressing it by 0.200m. The proposed solution is generally correct, but it is advised to use variables instead of numbers throughout the calculations to minimize errors. This approach allows for easier identification of mistakes during the problem-solving process. Overall, the emphasis is on accuracy and clarity in calculations for determining the spring constant.
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< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >[/color]

Could anyone please correct me if I have any mistake in the following question? It's very important for me to know if I am doing it correctly.

1)
"Michelle, a technician at a spring manufacturing plant,wants to make determining spring constants for fun. She fires a 0.0100kg bullet at a 500.0g wooden block that is attached to a massless spring;the bullet is traveling at 1000m/s when it strikes the block.The bullet lodges inside the block and the combination compresses the spring by 0.200m before it stops.Determine the spring constant that the technician oils calculate for this spring?"
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Solution:
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This looks very well done. My only recommendation is to use numbers more sparingly in your actual work. Use variables and plug in at the very end. It makes it easier to catch mistakes, both for you and for us.
 
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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