A biologist is studying growth in space. He wants to simulate Earth's gravitational field, so he positions the plants on a rotating platform in the spaceship. The distance of each plant from the central axis of rotation is r = 0.11 m. What angular speed is required?
To simulate Earth's gravitational field, the biologist needs to create a centrifugal force that is equivalent to the force of gravity on Earth.
Centrifugal force depends on the angular speed (ω), radius (r), and the mass of the object (m). In this case, we can assume the mass of the plants is constant for simplicity.
The equation for centrifugal force is:
Fc = m * ω^2 * r
We can equate this force to the force of gravity on Earth:
Fc = m * g
Where g is the acceleration due to gravity on Earth (approximately 9.8 m/s^2).
Setting these two equations equal to each other, we get:
m * ω^2 * r = m * g
The mass of the plants cancels out, so we're left with:
ω^2 * r = g
Solving for ω, we get:
ω = sqrt(g / r)
Now, we can substitute the known values into the equation:
g = 9.8 m/s^2 (acceleration due to gravity on Earth)
r = 0.11 m (distance of each plant from the central axis of rotation)
ω = sqrt(9.8 / 0.11)
ω ≈ sqrt(89.09)
ω ≈ 9.44 rad/s (rounded to two decimal places)
Therefore, the required angular speed to simulate Earth's gravitational field is approximately 9.44 rad/s.