A pendulum swings from the vertical through an angle of 15◦ on each side of the vertical...

Question

A pendulum swings from the vertical through an angle of 15◦ on each side of the vertical. If the pendulum is 90 cm long, what is the distance, x cm, between its highest and lowest points?

Answer 1

Hi Liza!

 

When the pendulum bob is at its lowest point, it is 90 cm below the rotation point, at the bottom of its swing.  When it is at its highest point, and its string (or whatever light body it swings on the end of) makes a 15º angle with the vertical, its vertical distance below its rotation point can be given by 

 

y = (90 cm) - (90 cm)cos(15º)

 

or

 

y = (90 cm)(1 - cos(15º))

 

If we call its horizontal position x = 0 at the lowest point, its horizontal displacement at its highest point is:

 

x = (90 cm)sin(15º)

 

To get the overall distance between the highest and lowest points, think of those x and y values as the sides of a right triangle, and the overall distance as the hypotenuse.  Use the Pythagorean theorem with x and y to get the overall distance between those two points.

 

I know this may be difficult to see without a diagram, so please feel free to let me know if you have any problems or questions in picturing what I am describing.  Good luck!

Answer 2

At its lowest point, the end of the pendulum is at Length L directly below the pivot point, where L is the length of the pendulum, which is 90 cm in this problem.  At its highest point, the vertical component of the free end of the pendulum is Lcosθ below the pivot point, where θ is the maximum angle the pendulum is displaced from the vertical, which is 15° in this problem.  Then the difference in heights is

 

Δh = L - Lcosθ = L(1 - cosθ) = (90 cm)(1 - cos15°) ≅ 3.07 cm

 

That is the difference in heights between its highest and lowest points.

Answer 3

Let the vertical position of the end of the pendulum =y

 

Let the horizontal position of the end of the pendulum=x

 

Then at the highest point, the distance x, from the center can be written as sin(90^o-15^o)=x/90, or

 

x=90*sin(75^o)

 

x=86.9 cm = the horizontal displacement.

 

To find y:

 

tan(15^o)=y/x, or,

 

y=86.9*tan(15^o)

 

y=23.3 cm = the vertical displacement of the pendulum.