Believe it or not, a plumb-bob does not hang exactly along a line directed to the centre of the Earth! Determine how far off true plumb a bob hangs in Cape Town. In which direction is the deviation (north, east, south, or west)? How would your answer differ for a plumb-bob in London? (Assume that the Earth is spherical.)
What is centripetal acceleration as a function of latitude?
w = 2 pi/24 hours = 2 pi /(24*3600) rad/s
Ac = w^2 r
r = Re cos (Lat)
where Re is earth radius and Lat is latitude angle
calculate Ac
call tension in plumb bob line T
then T will be off the direction to center of earth by angle A
T vector - mg vector = m Ac vector
draw those with mg toward earth center and Ac horizontal toward earth N-S axis
Then |T| cos A = m |Ac| cos Latitude +mg
and |T| sin A = m |Ac| sin Latitude
approximate |T| by m g
then sin A = (Ac/g) sin Latitude approximately
4.7 degrees
To determine how far off true plumb a bob hangs in Cape Town, we need to consider the concept of the Earth's curvature and the location of Cape Town on the Earth.
1. Earth's Curvature: Due to the Earth's shape, the force of gravity pulls objects towards its center. However, the Earth is not a perfect sphere, but rather an oblate spheroid, slightly flattened at the poles and bulging at the equator. This means that the gravitational pull is not perfectly perpendicular to the Earth's surface at all locations.
2. Location of Cape Town: Cape Town is located in the southern hemisphere, specifically in South Africa. This means that it is closer to the South Pole compared to the equator.
Considering these factors, we can determine the direction and magnitude of the deviation of a plumb bob in Cape Town:
- Direction of Deviation: Since Cape Town is closer to the South Pole, the deviation of a plumb bob will be towards the pole. Therefore, the direction of deviation is towards the south.
- Magnitude of Deviation: The magnitude of the deviation depends on the latitude of the location. Cape Town is roughly around 33°57′00″S latitude. At this latitude, the deviation from true plumb is estimated to be around 0.53 arcminutes or approximately 19.5 arcseconds.
Now, let's consider the case of a plumb-bob in London:
- Location of London: London is in the northern hemisphere, further away from the South Pole compared to Cape Town.
The deviation of a plumb-bob in London would be in the opposite direction compared to Cape Town. Therefore, the direction of deviation for a plumb-bob in London would be towards the north.
The magnitude of deviation would be similar to Cape Town's case, with a roughly estimated deviation of 0.53 arcminutes or approximately 19.5 arcseconds.
So, to summarize: The deviation of a plumb-bob in Cape Town is towards the south, while the deviation in London would be towards the north. The magnitude of deviation is approximately 0.53 arcminutes or 19.5 arcseconds in both cases, assuming a spherical Earth.