SHM: Different Final Answers When Using Energy Conversation & Trigonometry

[Oz Alikhan]

1. The Question from Text:

"A forest playground has a tyre hanging from a tree branch. The tyre behaves like a pendulum, with a rope of 4.0 metres length, and the tyres mass 15 kg. A child of mass 45 kg swings on the tyre by pulling it 3.0 metres to one side and leaping on. What is the maximum height that the tyre will reach above its equilibrium position?" 3. The Solutions:

Through Energy Conservation

T = 2∏ \sqrt{l/g} ≈4.0s
ω = 2∏/T ≈1.6 rad/s

Therefore, V_max = Aω = 3.0 x 1.6 = 4.7 (Using unrounded T & ω)

Max K.E = 0.5*(15+45)*(4.7)^2 = 660 J

Max P.E = 660 J = mgh
h = 660/mg = 1.1 mThrough Trigonometry

The right angled triangle: Cosθ = 3/4
If I use Pythagoras Theorem, I get \sqrt{4^{2} - 3^{2}} = \sqrt{7}
Then, 4 - \sqrt{7} = 1.35 mWhy is there a significant difference in my answers? (Quick side note: Why doesn't my MathTex not work?)

 

[gneill]

1. The Question from Text:

"A forest playground has a tyre hanging from a tree branch. The tyre behaves like a pendulum, with a rope of 4.0 metres length, and the tyres mass 15 kg. A child of mass 45 kg swings on the tyre by pulling it 3.0 metres to one side and leaping on. What is the maximum height that the tyre will reach above its equilibrium position?"


3. The Solutions:

Through Energy Conservation

T = 2∏ \sqrt{l/g} ≈4.0s
ω = 2∏/T ≈1.6 rad/s

Therefore, V_max = Aω = 3.0 x 1.6 = 4.7 (Using unrounded T & ω)

Max K.E = 0.5*(15+45)*(4.7)^2 = 660 J

Max P.E = 660 J = mgh
h = 660/mg = 1.1 m


Through Trigonometry

The right angled triangle: Cosθ = 3/4
If I use Pythagoras Theorem, I get \sqrt{4^{2} - 3^{2}} = \sqrt{7}
Then, 4 - \sqrt{7} = 1.35 m


Why is there a significant difference in my answers? (Quick side note: Why doesn't my MathTex not work?)

A pendulum is not a "good" SHM device except for small displacement angles (where "small" means sin(θ) ≈ θ). Here your angle is nearly 50°, so expect large inaccuracies!

MathTex expressions should be between appropriate tags that flag MathTex to interpret the enclosed text. For expression embedded in text you can use [ itex ] and [ /itex ] (no spaces) or a pair of ## . Larger, one expression per line version uses tags[ tex ] ... [ /tex ] (again no spaces) or a pair of $$'s

 

[Oz Alikhan]

Thanks for the reply. That is what I thought, although I tried to improve this approximation by using the formula that takes into account bigger amplitudes: http://upload.wikimedia.org/wikipedia/en/math/3/0/1/3014cdccf9d3df9770a001b84ae6ac80.png

However, the percentage error difference between my two answers even got bigger after I used that formula.

 

[Oz Alikhan]

Oh I see. The Trigonometry method is a linear approximation that does not take into account of the larger angles where the approximation deviates from this linearity. That explains the difference in answers .

Thanks for clearing that up

 

[asteriatic]

I would like to point out that both solutions are correct and valid, but they are using different methods to calculate the maximum height of the tyre. The first solution uses the principle of energy conservation, which considers the initial and final energy of the system. The second solution uses trigonometry to calculate the height of the tyre based on the angle at which the child pulls the tyre.

The difference in the answers can be attributed to the fact that the two methods are based on different assumptions and equations. The energy conservation method assumes that all the energy is converted from kinetic to potential at the highest point, while the trigonometry method assumes that the height of the tyre at the highest point is equal to the vertical displacement of the child.

It is important to note that both methods have their own limitations and may not give the exact same answer. It is also worth mentioning that the difference in the answers may be due to rounding in the calculations.

As for the issue with MathTex, it is likely a technical issue and I would suggest seeking help from a technical support team. In the future, it would be helpful to provide more context and information about the issue to better address it.