- #1

mopit_011

- 17

- 8

- Homework Statement:
- Because of the rotation of the Earth, a plumb bob may not hang exactly along the direction of the Earth’s gravitational force on the plumb bob but may deviate slightly from that direction. (a) Show that the deflection theta in radians at a point is equivalent to ##\theta = (2\pi^2R/gT^2)(\sin 2L)## where L is the latitude at the point, R is the radius, and T is the period of the Earth’s rotation.

- Relevant Equations:
- Net force in uniform circular motion = (mv^2/r).

I started by making my coordinate system so that the x-axis aligned with the radius of the circle at a certain latitude L and the positive direction was facing away from the center of the circle, and the y-axis was parallel to the vertical axis of the Earth. Then, I wrote the equations for the net forces for both axes. Here are the equations I wrote:

(1) ##F_{x}## = ##T \cos (L + \theta) ## ##- mg## ##\cos(L)## ##= (mv^2/R \cos L)## where R is the radius of the Earth.

(2) ##F_{y} = T \sin (L + \theta) - mg \sin (L)##.

Then I solved for T from the second equation and plugged it into the first equation.

When I tried to solve for theta, I was unable to show the desired deflection angle was correct. Are my equations wrong or am I missing something else? Thank you!

(1) ##F_{x}## = ##T \cos (L + \theta) ## ##- mg## ##\cos(L)## ##= (mv^2/R \cos L)## where R is the radius of the Earth.

(2) ##F_{y} = T \sin (L + \theta) - mg \sin (L)##.

Then I solved for T from the second equation and plugged it into the first equation.

When I tried to solve for theta, I was unable to show the desired deflection angle was correct. Are my equations wrong or am I missing something else? Thank you!