Determine the magnitude of the centripetal acceleration of an object on the surface of the Earth at a latitude of 71.6°. Consider the Earth to be a sphere of radius
R = 6.38 ✕ 106 m.
Correction: 6.38x10^6 m
r = 6.38 * 10^6 * cos 71.6
= 2.01 * 10^6 meters
angular velocity = w
= 2 pi radians/24 hours
*1 hour/3600 seconds
= 2 pi / (24*3600)
= 7.27 * 10^-5 radians/second
so
Ac = r w^2
= 2.01*10^6 (7.27)^2*10^-10 m/s^2
To determine the magnitude of the centripetal acceleration of an object on the surface of the Earth at a given latitude, we need to consider two factors: the rotational speed of the Earth and the radius of the Earth at that latitude.
Step 1: Find the rotational speed of the Earth.
The Earth rotates once every 24 hours. The rotational speed, v, can be calculated using the formula:
v = (2πR)/T,
where R is the radius of the Earth and T is the period of rotation (24 hours or 86,400 seconds).
Using the given radius R = 6.38 × 10^6 m and the period T = 24 hours = 86,400 seconds, we can substitute these values into the formula to find the rotational speed of the Earth.
v = (2π × 6.38 × 10^6 m) / 86,400 s.
Calculating this expression will give us the value of v.
Step 2: Convert the latitude to radians.
To find the radius of the Earth at a given latitude, we need to convert the latitude from degrees to radians. The conversion formula is:
θ (radians) = θ (degrees) × (π/180),
where θ is the angle in radians.
Using the given latitude of 71.6°, we can substitute this value into the formula to find the angle in radians.
θ (radians) = 71.6° × (π / 180).
Calculating this expression will give us the value of θ in radians.
Step 3: Find the radius of the Earth at the given latitude.
The radius of the Earth at a given latitude can be calculated using the formula:
R' = R × cos(θ),
where R is the radius of the Earth and θ is the latitude in radians.
Using the given radius of the Earth R = 6.38 × 10^6 m and the angle in radians θ, we can substitute these values into the formula to find the radius of the Earth at the given latitude.
R' = 6.38 × 10^6 m × cos(θ).
Calculating this expression will give us the value of R' in meters.
Step 4: Calculate the centripetal acceleration.
The centripetal acceleration, ac, is given by the formula:
ac = v^2 / R',
where v is the rotational speed of the Earth and R' is the radius of the Earth at the given latitude.
Using the calculated values for v and R' from steps 1 and 3, we can substitute these values into the formula to find the magnitude of the centripetal acceleration.
ac = (v^2) / R'.
Calculating this expression will give us the magnitude of the centripetal acceleration at the given latitude.