At ground level g is 9.8m/s^2. Suppose the earth started to increase 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 ?
To determine how an increase in the Earth's angular velocity would affect the length of a day, we need to understand the relationship between angular velocity, rotational motion, and the concept of centripetal force.
The Earth's rotational motion causes a centripetal force that keeps objects, including people, on its surface. The centripetal force is provided by the gravitational force, which is proportional to the mass of an object and the acceleration due to gravity (g).
When the angular velocity of the Earth increases, the centripetal force required to keep objects on its surface also increases. To calculate the new length of a day, we need to find the angular velocity at which people would be "thrown off" the Earth (not physically possible due to gravity but hypothetically).
To start, we need to find the critical angular velocity (ω) at which people would no longer experience the centripetal force provided by gravity. We can equate this centripetal force with the gravitational force:
m * ω^2 * R = m * g
Here, m represents the mass of a person, ω is the angular velocity, R is the radius of the Earth, and g is the acceleration due to gravity at the Earth's surface.
The mass of a person cancels out from both sides of the equation:
ω^2 * R = g
Next, we can rearrange the equation to solve for ω:
ω = √(g / R)
Given that g is approximately 9.8 m/s^2 and the radius of the Earth (R) is about 6,371 km or 6,371,000 meters, we can calculate ω:
ω = √(9.8 / 6,371,000)
Once we have the value of ω, we can relate it to the length of a day. The length of a day, or the time it takes for the Earth to complete one rotation, is inversely proportional to the angular velocity. Therefore, as the angular velocity (ω) increases, the length of a day decreases.
To calculate the new length of a day, we need to compare ω with the current angular velocity of the Earth:
ω_new / ω_earth = T_earth / T_new
Where ω_new is the angular velocity at which people would be hypothetically "thrown off," ω_earth is the current angular velocity of the Earth, and T_earth and T_new are the current and new lengths of a day, respectively.
Let's say a day on Earth is currently 24 hours or 86,400 seconds (T_earth = 86,400 seconds). We can rearrange the equation to solve for T_new:
T_new = (ω_earth / ω_new) * T_earth
Using the calculated value of ω from earlier and assuming the current angular velocity of the Earth (ω_earth) is approximately 7.27 x 10^(-5) rad/s, we can calculate T_new.
Finally, it's important to note that the expression "thrown off" is not accurate because people would not be thrown off the Earth due to an increase in angular velocity. The force of gravity would still act on them, even if the rotation speed increased. Instead, a more appropriate term to use would be experiencing a reduced gravitational pull that would affect their ability to stay in their normal positions on the surface.