# Arc of a pendulum (Trigonometry)

How far does the tip of a 5-foot pendulum travel as it swings through an angle of 30 degrees? Round your answer to two decimal places.

I proceeded by drawing a picture of my swinging pendulum with dimensions. enter image description here

I drew a right triangle next to my pendulum with the length of my pendulum (5') being the x-axis of my right triangle. My 30 degree theta angle is my tangent angle. I solved for tangent and got 2.89 for my y-axes. I used the Pythagorean theorem to solve for the length of my hypotenuse which would be my answer C=5.77'. I ask the community if my steps are sound in judgement.

• Hint: $s=r\theta$ where $\theta$ is in radians. – John Douma Aug 28 '18 at 20:55
• You don't need to use the Pythagorean theorem to measure an arc length. You have the wrong picture for your problem. – Doug M Aug 28 '18 at 21:03
• @CuriousGeorge Are you looking for the horizontal distance travelled by the tip or to the arc length? – gimusi Aug 28 '18 at 21:14
• You have two triangles drawn on the paper you were working on. One of them is drawn so that one of its legs (the one next to the 30 degree angle) lies along the arm of the pendulum, you have marked the distance "5'" next to the pendulum and it appears to me that the "5'" line next to the pendulum is much longer than the leg of the triangle next to the 30 degree angle. Yet in the other triangle drawing you say the leg next to the 30 degree angle is 5'. It looks like two different size triangles to me. – David K Aug 28 '18 at 21:34
• It is not clear how you think the hypotenuse of the second triangle relates to the question. – David K Aug 28 '18 at 21:34

You want the arc length of an arc spanning $30°$ with radius $r= 5.0$

1. Convert the angle into radians
2. Use the arc length formula $s = r\, \theta$

Thirty degrees is one-twelfth of a complete circle (since a complete circle is $360$ degrees). The circumference of a circle of radius $R$ is $2\pi R$. So if a $5$-foot pendulum swings through an angle of $30$ degrees, its tip travels an arc-length distance of $(2\pi\times5)/12=5\pi/6\approx2.618$ feet, which can be rounded up to $2.62$ feet.

If by "distance traveled" you mean the linear distance between the two endpoints of the tip's $30$-degree swing, then the answer is

$$5\sqrt{2-2\cos(30^\circ)}=5\sqrt{2-\sqrt3}\approx2.58819\approx2.59\text{ feet}$$

We have that the lenght of the pendulum is constant therefore the distance in horizontal direction is

$$d=H \cdot \sin 30° = 5 \cdot \frac12 = 2.50$$

and the arc length

$$s=R\cdot \theta=5\cdot \frac{30\pi}{180}\approx 2.62$$

• @DougM I've given the distance travelled horizontally! I fix that and add also the arc length. Thanks – gimusi Aug 28 '18 at 21:03
• The distance traveled horizontally is $5\cdot 2 \cdot \sin 15^\circ.$ And the question of how far does the tip travel is $\frac {5\pi}{6}$ – Doug M Aug 28 '18 at 21:07
• @DougM It was not so clear to me reading the OP and alsofrom the linked image the sketch represents a swing of 30° such that the horizontal distance travelled is equal to $5\cdot \sin 30°$. – gimusi Aug 28 '18 at 21:12