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. (a) Assuming the earth is a sphere with a radius of 6.38 106 m, determine the speed of a person situated at the equator.
(b) What is the speed of a person situated at a latitude of 29.0° north of the equator?
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(a) To determine the speed of a person situated at the equator, we need to calculate the linear velocity of a point on the equator due to the Earth's rotation. The formula for linear velocity is given by:
v = rω,
where v is the linear velocity, r is the radius of the Earth, and ω is the angular velocity or the rate at which the Earth rotates.
The angular velocity of the Earth can be calculated by the formula:
ω = 2π/T,
where T is the period of rotation, which is equal to 24 hours or 86,400 seconds (since the Earth rotates once per day).
Substituting the values into the equations:
ω = 2π/86,400 s ≈ 7.27 × 10^(-5) rad/s,
r = 6.38 × 10^6 m.
Now we can find the linear velocity v:
v = rω
= (6.38 × 10^6 m)(7.27 × 10^(-5) rad/s)
≈ 464.5 m/s.
Therefore, the speed of a person situated at the equator is approximately 464.5 m/s.
(b) To determine the speed of a person situated at a latitude of 29.0° north of the equator, we need to take into account the fact that the radius of the Earth at this latitude is slightly smaller than at the equator. We can use the formula:
v' = r'ω,
where v' is the linear velocity at the given latitude, r' is the radius of the Earth at that latitude, and ω is the angular velocity.
The radius at a given latitude can be calculated by the formula:
r' = r * cos(θ),
where r is the radius of the Earth and θ is the latitude in radians.
Converting the latitude to radians:
θ = 29.0° * (π/180)
≈ 0.506 rad.
Substituting the values into the equation:
r' = (6.38 × 10^6 m) * cos(0.506 rad).
Calculate the cos(0.506 rad) using a calculator:
cos(0.506 rad) ≈ 0.8776.
Now we can find the linear velocity v':
v' = r'ω
≈ (6.38 × 10^6 m) * 0.8776 * 7.27 × 10^(-5) rad/s
≈ 381.7 m/s.
Therefore, the speed of a person situated at a latitude of 29.0° north of the equator is approximately 381.7 m/s.
To determine the speed of a person situated at the equator, we first need to find the distance traveled by a point on the equator in one day. This distance is equal to the circumference of the Earth.
(a) The circumference of a sphere can be calculated using the formula:
C = 2πr
where C is the circumference and r is the radius of the sphere. Given that the radius of the Earth is 6.38 × 10^6 m, we can substitute the value into the formula:
C = 2π(6.38 × 10^6 m)
= 2 × 3.14159 × (6.38 × 10^6 m)
= 40.07396 × 10^6 m
Therefore, the distance traveled by a person at the equator in one day is approximately 40.07 × 10^6 meters.
To find the speed at the equator, we divide the distance traveled by the time taken, which is one day (24 hours or 86,400 seconds).
Speed = Distance / Time
= 40.07 × 10^6 m / 86,400 s
= 463.01 m/s
So, a person situated at the equator is moving at a speed of approximately 463.01 meters per second.
(b) To determine the speed of a person situated at a latitude of 29.0° north of the equator, we need to consider the effect of the Earth's rotation on the circumference at that latitude.
At a latitude other than the equator, the distance traveled by a point on the Earth's surface in one day is given by the equation:
C' = 2πr'cos(θ)
where C' is the distance traveled, r' is the radius of the latitude circle (i.e., the distance from the Earth's axis to the latitude), and θ is the latitude.
Given that the latitude is 29.0°, which is north of the equator, we need to calculate the radius of the latitude circle.
The radius of the latitude circle can be calculated using the formula:
r' = r * cos(θ)
where r is the radius of the Earth and θ is the latitude. Substituting the values:
r' = (6.38 × 10^6 m) * cos(29.0°)
= (6.38 × 10^6 m) * cos(0.506 rad)
≈ 5.61 × 10^6 m
Now we can substitute this value and the latitude into the equation for the circumference at the given latitude:
C' = 2π(5.61 × 10^6 m) * cos(29.0°)
≈ 35.30 × 10^6 m
To find the speed at this latitude, we divide the distance traveled by the time taken, which is one day (24 hours or 86,400 seconds).
Speed = Distance / Time
= 35.30 × 10^6 m / 86,400 s
= 408.56 m/s
Therefore, a person situated at a latitude of 29.0° north of the equator is moving at a speed of approximately 408.56 meters per second.