Vector Resolution and Force Balancing on a Force Table

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To solve for the resultant and opposing forces on a force table, it is essential to break down the vectors into their horizontal and vertical components. Each vector can be represented as the hypotenuse of a right triangle, allowing for the calculation of the other two sides using trigonometric functions. Drawing a diagram can aid in visualizing the angles and components, making it easier to understand the relationships between them. For the given angles and masses, applying these principles will help achieve balance on the force table. Understanding vector resolution is crucial for accurately determining the necessary forces and angles.
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I need some help with the following assignment!

For the following sets of angles and masses, you must solve for the resultant force, the opposing force, and the angle and mass needed to create a balance on a force table.

1. 50 g at 20 degrees, 100 g at 65 degrees
2. 50g at 90 degrees, 50g at 270 degrees
 
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I am not sure where to start with figureing these out because I have a lot of trouble with vectors and understanding them.
 
well, is there any other way you can think of these vectors that would make them simpler?

what if we broke them up into parts?
 
Well see that is where I have the problem. I do not understand how to break them up.
 
ok

we want to describe the vector with a horizontal and vertical component instead of one part with a direction

so what we basically have is the hypotenuse of a right angle triangle, and we want to find the other two sides

well draw yourself a diagram to help you understand it

we have all the angles and a side, so we can find the rest of the triangle
 
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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