What is the phase constant in SHM?

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The phase constant \(\phi\) in simple harmonic motion (SHM) is determined from the equation \(x(t) = A \cos(wt + \phi)\), where \(A\) is the amplitude and \(w\) is the angular frequency. To find \(\phi\), one can use the initial condition \(x(0) = A \cos(\phi)\), leading to the formula \(\phi = \cos^{-1}(x(0)/A)\). Alternatively, if the position \(x(t)\) is known at a specific time \(t\), \(\phi\) can be calculated using \(\phi = \cos^{-1}(x(t)/A) - wt\). The discussion highlights that for a specific scenario, this calculation results in a phase constant of \(0.5\pi\). Understanding the phase constant is crucial for analyzing oscillatory motion in SHM.
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The graph shows the position x of an oscillating object as a function of time t. The equation of the graph is x(t)=Acos(wt + \phi)
where A is the amplitude, w is the angular frequency, and \phi is a phase constant. The quantities M,N, and T are measurements to be used in your answers.

See attached image.

What is \phi in the equation?
 

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can anyone help me?
 
The attachment has to be approved, but the phase angle (constant) is found for example when t = 0, and knowing x(t=0) = A cos \phi, or

\phi = cos-1 (x(0)/A), or

if x(t) is known at t, then

\phi = cos-1 (x(t)/A) - wt
 
the answer leads to 0.5pi.
 
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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