Show that Newton's 2nd Law is *Not* valid in a reference frame

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Newton's Second Law is not valid in a reference frame that is accelerating relative to a laboratory frame, as the relationship between force, mass, and acceleration breaks down. In a non-inertial frame, the observer would measure forces differently, leading to discrepancies in the expected results of F = ma. The discussion emphasizes that while the law holds in an inertial frame, it fails when the frame itself is accelerating. An effective approach to demonstrate this is to show that the force acting on an object does not equal its mass times its acceleration in the accelerating frame. Thus, the validity of Newton's Second Law is contingent on the choice of reference frame.
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Homework Statement


Show that Newton's Second Law is NOT valid in a reference frame moving past the laboratory frame of problem 1 with a constant acceleration?
Problem 1: In a laboratory frame of reference, an observer notes that Newton's Second Law is valid. Show that it is also valid for an observer moving at a constant speed, small compared with the speed of light, relative to the laboratory frame.
dx^1/dt = dx/dt-v
d^2x^1/dt^2= d^2x/dt^2
force new = force old
acceleration = d^2x/dt^2 and if v is constant = dv/dt = 0

Homework Equations


F = ma

The Attempt at a Solution


I am not sure how to solve it
 
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You don't have to be sure of the approach to use the approach.
Try showing that the force on the object is not equal to it's mass multiplied by it's acceleration.
 
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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