A plumb bob does not hang exactly along a line directed to the center of the Earth's rotation. How much does the plumb bob deviate from a radial line at 38.5° north latitude?
I got my answer to be 0.122288439 deg but it was incorrect. I used theta= a(sin 38.5)/g (which produces an answer in radians and I converted to degrees)Please & thank you!
You may have used a wrong value for a, the centripetal acceleration at the equator. Also, there should be a cosine of latitude, not sine, in the formula. There is no rotation effect at the poles, where your formula woulde say it it a maximum.
a = R w^2, where R is the radius of the earth and w is the angular rotation rate in rad/s.
Approximate values are:
R = 6.4*10^6 m
w = 7.27*10^-5 rad/s
a = Rw^2 = 3.4*10^-2 m/s^2
a/g*cos 38.5 = 2.8*10^-3 radians
= 0.16 degrees
You also have to many significant figures in your answer.
To determine the deviation of a plumb bob from a radial line at a given latitude, you can use the concept of the Earth's rotation and its effects on gravity. Here's the correct method to calculate the deviation:
1. Start by considering the forces acting on the plumb bob. The primary force is gravity, acting vertically downward.
2. Due to the Earth's rotation, there will be a centrifugal force acting horizontally away from the Earth's axis. This force counteracts gravity.
3. The deviation of the plumb bob occurs because the net force acting on it is not purely vertical. It will have a small horizontal component toward or away from the Earth's axis.
4. Let's denote the deviation angle as Δθ. By considering the forces in the system, we can express this angle mathematically.
5. The centrifugal force acting on the plumb bob depends on the latitude. Let's denote the Earth's angular velocity as ω and the radius at the given latitude as r.
6. The centrifugal force is given by Fc = mω^2r, where m represents the mass of the plumb bob.
7. The gravitational force acting on the plumb bob is given by Fg = mg, where g represents the acceleration due to gravity.
8. Now, using trigonometry, we can find the deviation angle. We know that sin(Δθ) = Fc / Fg.
9. Substituting the expressions for Fc and Fg, we get sin(Δθ) = (mω^2r) / (mg).
10. The masses, m, cancel out, so we are left with sin(Δθ) = ω^2r / g.
11. Finally, solve for Δθ by taking the arcsine of both sides: Δθ = arcsin(ω^2r / g).
12. Make sure to use the correct units for ω, r, and g to ensure consistent results.
Using this method, you should be able to obtain the accurate deviation angle. If you have the values of ω, r, and g, you can plug them into the equation to find the correct answer.