at ground level g is 9.8m/s^2 suppose the earth starts to increas its angular velocity how long would a day be when people on the equator were just thrown off ?why is the expression thrown off a bad one ?
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To determine the change in the length of a day when people on the equator would be thrown off due to an increase in the Earth's angular velocity, we need to consider the relationship between the Earth's rotation and the gravitational force.
First, let's understand the concept of centrifugal force. When an object rotates around a central axis, an outward force acts on objects along the radius called centrifugal force. This force increases with the square of the angular velocity and the distance from the axis of rotation.
To calculate the time it would take for people on the equator to be thrown off, we need to find the point at which the centrifugal force matches the force of gravity. At this point, the gravitational force will no longer hold people on the Earth's surface.
Now, let's calculate:
1. Determine the gravitational force at the equator:
The force of gravity experienced at the surface of the Earth is given by the formula: F = m * g, where g is the acceleration due to gravity. Given g = 9.8 m/s^2, we can use this value for our calculations.
2. Calculate the centrifugal force at the equator:
The centrifugal force is given by the formula: F = m * ω^2 * r, where ω is the angular velocity and r is the radius of the Earth. Since the Earth is approximately spherical, we can take the average radius, which is about 6,371 km (or 6,371,000 meters).
Let's assume the initial angular velocity of the Earth is ω₁, and we want to find the final angular velocity where people would be thrown off, which we'll call ω₂.
At the point where people would be thrown off, the centrifugal force would match the gravitational force: m * ω₂^2 * r = m * g.
Since mass (m) is common to both sides of the equation, we can cancel it. Therefore, we can write: ω₂^2 * r = g.
3. Solve for the final angular velocity (ω₂):
Rearrange the equation to solve for ω₂: ω₂ = √(g / r).
Plug in the given values of g = 9.8 m/s^2 and r = 6,371,000 meters into the equation to find ω₂.
4. Determine the change in the length of a day:
The length of a day is directly proportional to the angular velocity of the Earth. Thus, we can say that the change in the length of a day is proportional to the change in the Earth's angular velocity.
For example, if the initial length of a day is 24 hours (or 86,400 seconds) and the initial angular velocity is ω₁, then the final length of a day, given the final angular velocity ω₂, can be calculated using the formula: Final_Day = Initial_Day * (ω₁ / ω₂).
Plug in the values of the initial day and the calculated ω₂ to find the final length of a day.
Regarding the expression "thrown off," it's not an accurate way to describe what would happen. The force causing people to lose contact with the Earth's surface would be the centrifugal force exceeding the gravitational force, rather than being forcefully thrown off. This is because the gravitational force at that point can no longer counterbalance the centrifugal force acting outwards.