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# Pendulum Trigonometry

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Hello

Here's a tricky one that I could use an explanation and answer to:

When a pendulum swings 40 degrees from the vertical, the bob moves 20 cm horizontally and 7.3 cm vertically. What is the length of the pendulum to the nearest centimeter?

Thanks!

Jedithious  Jul 2, 2014

#4
+26640
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A picture helps:

1.  sin(θ) = 20/L  ;  L = 20/  sin(θ);    θ = 40°

$${\mathtt{L}} = {\frac{{\mathtt{20}}}{\underset{\,\,\,\,^{{360^\circ}}}{{sin}}{\left({\mathtt{40}}^\circ\right)}}} \Rightarrow {\mathtt{L}} = {\mathtt{31.114\: \!476\: \!537\: \!185\: \!947\: \!2}}$$

L = 31 cm to nearest cm

2. Lcos(θ) + 7.3 = L;  L = 7.3/(1 - cos(θ));

$${\mathtt{L}} = {\frac{{\mathtt{7.3}}}{\left({\mathtt{1}}{\mathtt{\,-\,}}\underset{\,\,\,\,^{{360^\circ}}}{{cos}}{\left({\mathtt{40}}^\circ\right)}\right)}} \Rightarrow {\mathtt{L}} = {\mathtt{31.202\: \!507\: \!422\: \!010\: \!490\: \!6}}$$

L = 31 cm to nearest cm.

Alan  Jul 2, 2014
Sort:

#1
+85807
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The pendulum is 31.114476537214339461859 cm. long

To see this, note that when the pedulum swings 40 degrees from the vertical, the horizontal component is now 20cm. So, to find the vertical component in this position we have

20/tan 40 = 23.8350718518921531 cm long

So the total length of the  pedulum is just √(202 + 23.83507185189215312) = 31.114476537214339461859cm

To see that this is true, note that when it swings back to vertical, the vertical component changes from

23.8350718518921531 cm to 31.114476537214339461859 cm (the full pendulum length, there is no horizontal component in this position!!!)

So the difference in this length minus the  the vertical component length when the pendulum is swung 40 degrees from the horizontal is just 31.114476537214339461859 cm - 23.8350718518921531 cm =

7.279404685322186361859 cm or about 7.3 cm !!

CPhill  Jul 2, 2014
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$$\\ \tan{(40\ensuremath{^\circ})}={20\;cm \over l-7.3\;cm}\\\\ (l-7.3\;cm)={20\;cm \over \tan{(40\ensuremath{^\circ})} }\\\\ l=7.3\;cm+{20\;cm \over \tan{(40\ensuremath{^\circ})} }\\\\ l=7.3\;cm + 23.8\;cm=31.1\;cm$$

The pendulum is 31.1 cm long

heureka  Jul 2, 2014
#3
+10

As an alternative method of solution, resolving the displaced pendulum to the vertical and adding the 7.3 gets you the equation

L*cos(40) + 7.3 = L, (the 20 isn't needed)

and from which L = 31.2025 (4 dp).

It would seem that we can't be sure of the first decimal place and that the best that we can do is to give the result correct to the nearest cm, as asked for in the question.

Guest Jul 2, 2014
#4
+26640
+13

A picture helps:

1.  sin(θ) = 20/L  ;  L = 20/  sin(θ);    θ = 40°

$${\mathtt{L}} = {\frac{{\mathtt{20}}}{\underset{\,\,\,\,^{{360^\circ}}}{{sin}}{\left({\mathtt{40}}^\circ\right)}}} \Rightarrow {\mathtt{L}} = {\mathtt{31.114\: \!476\: \!537\: \!185\: \!947\: \!2}}$$

L = 31 cm to nearest cm

2. Lcos(θ) + 7.3 = L;  L = 7.3/(1 - cos(θ));

$${\mathtt{L}} = {\frac{{\mathtt{7.3}}}{\left({\mathtt{1}}{\mathtt{\,-\,}}\underset{\,\,\,\,^{{360^\circ}}}{{cos}}{\left({\mathtt{40}}^\circ\right)}\right)}} \Rightarrow {\mathtt{L}} = {\mathtt{31.202\: \!507\: \!422\: \!010\: \!490\: \!6}}$$

L = 31 cm to nearest cm.

Alan  Jul 2, 2014
#5
+85807
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We seem to have a "variable length" pendulum, Jedithious !!!

We'll use 31cm and call it good.......(Sorry, I forgot that the question requested to the nearest cm!!...my bad)

CPhill  Jul 2, 2014
#6
+154
+5

CPhil, heureka, anonymous, and Alan: Thank you very much for taking the time to help out!

Jedithious  Jul 5, 2014

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