How Does Trigonometry Determine the Height in a Ballistic Pendulum?

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SUMMARY

The height \( h \) that a ballistic pendulum rises after capturing a ball is determined by the equation \( h = R(1 - \cos \theta) \), where \( R \) represents the length of the pendulum and \( \theta \) is the maximum deflection angle. The discussion clarifies that for the height to increase, \( \cos(\theta) \) must yield a value less than 1, ensuring that \( h \) remains positive. The participant initially struggled with the concept but successfully deduced the solution independently, demonstrating a solid understanding of the relationship between trigonometric functions and pendulum motion.

PREREQUISITES
  • Understanding of basic trigonometric functions, specifically cosine.
  • Familiarity with the principles of pendulum motion.
  • Knowledge of the relationship between angle and height in physics.
  • Ability to interpret mathematical equations related to physical phenomena.
NEXT STEPS
  • Study the derivation of the ballistic pendulum equations in detail.
  • Explore the implications of maximum deflection angles on pendulum dynamics.
  • Learn about energy conservation principles in pendulum systems.
  • Investigate real-world applications of ballistic pendulums in physics experiments.
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and trigonometry, as well as educators looking for practical examples of pendulum motion and energy conservation principles.

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Homework Statement


Using trigonometry and Fig. 2 in the ballistic pendulum write-up, show that the height h the pendulum rises after capturing the ball is given by: h=R(1-cosθ)

Figure two is here: http://imgur.com/W9nvVZT


Homework Equations



h=R(1-cosθ)

The Attempt at a Solution


So in order for the height to increase I understand that R is the length of the pendulum. θ is the maximum deflection, so since R can't be negative that means cos(θ) must return a negative answer in order for the height to increase. So how do I prove that cos(θ) in this instance will be a negative number, thus making the height increase.
 
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I'm an idiot and figured this question out the second I hit submit, so nevermind I got it! Not sure how to delete a post or if I just let it go!
 
If you had been an idiot, you wouldn't have found it out by yourself, nor would the best teacher in the world (not me!) been able to make you understand.
So, conclusion:
You are not an idiot after all! :smile:
 

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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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