The earth rotates once per day about an axis passing through the north and south poles, an axis that is perpendicular to the plane of the equator. Assuming the earth is a sphere with a radius of 6.38 106 m, determine the speed and centripetal acceleration of a person situated at each of the following.
(a) the equator
1 m/s
2 m/s2
(b) a latitude of 42.0° north of the equator
3 m/s
4 m/s2
(a) ω=2π/24•3600 rad/s
v= ω•R=...
a=ω²R=…
(b)
r=R•cos42°
v1= ω•r=...
a1=ω²r=…
To determine the speed and centripetal acceleration of a person situated at different locations on Earth, we need to consider the rotational motion of the Earth.
(a) At the equator:
The speed of a person at the equator can be calculated using the formula:
Speed = radius × angular velocity
The angular velocity of the Earth can be calculated using the period of rotation, which is 24 hours or 86,400 seconds.
Angular velocity = (2π radians) / (time period in seconds)
Angular velocity = (2π radians) / (86,400 seconds)
Substituting the values:
Angular velocity = (2π) / (86,400 seconds)
The radius of the Earth is given as 6.38 × 10^6 m:
Speed = (6.38 × 10^6 m) × [(2π) / (86,400 seconds)]
Calculating this value will give us the speed of a person at the equator.
To calculate the centripetal acceleration at the equator, we can use the formula:
Centripetal acceleration = (Speed)^2 / Radius
Substituting the values:
Centripetal acceleration = (Speed at the equator)^2 / (Radius of the Earth)
Calculating this value will give us the centripetal acceleration at the equator.
(b) At a latitude of 42.0° north of the equator:
For any latitude other than the equator, the radius of the circle swept by the rotation decreases as you move away from the equator. The radius of rotation at this latitude would be smaller than the radius of the Earth.
To calculate the speed at this latitude, we need to find the radius of this smaller circle. This can be done using the formula:
Radius of circle = (Radius of the Earth) × cosine (latitude)
Substituting the values:
Radius at 42.0° north of the equator = (6.38 × 10^6 m) × cosine (42.0°)
The speed can be calculated using the same formula as before:
Speed = (Radius at 42.0° north of the equator) × [(2π) / (86,400 seconds)]
To calculate the centripetal acceleration, we can again use the formula:
Centripetal acceleration = (Speed at 42.0° north of the equator)^2 / (Radius at 42.0° north of the equator)
Calculating these values will give us the speed and centripetal acceleration at a latitude of 42.0° north of the equator.
Please feel free to plug in the numbers and calculate the final results.