Understanding the Right Hand Rule for Rotational Dynamics

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The right hand rule is a crucial tool for understanding rotational dynamics, particularly in solving torque problems. To apply it, one points the thumb of the right hand in the direction of the rotational axis, while the fingers curl in the direction of rotation. Positive torque is indicated when the rotation follows the direction of the curled fingers, while negative torque occurs when it goes against that direction. This rule helps in determining the direction of angular momentum and angular velocity as well. Mastering the right hand rule is essential for accurately analyzing rotational motion in physics.
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I don't get the right hand rule that apply to rotational dynamics. How could I use to solve a torque problem and how to know when its positive and what is negative.
 
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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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