To tackle the problem of an astronaut being tested in a centrifuge, we need to break down the information given and apply some physics concepts related to rotational motion. The centrifuge has a radius of 10.4 meters and the angular position as a function of time is given by the equation ϕ = (0.326 rad/s²)t². Let's go step by step to find the astronaut's angular speed, tangential speed, tangential acceleration, and radial acceleration at t = 5.60 seconds.
Angular Speed Calculation
First, we need to find the angular speed (ω) of the centrifuge. Angular speed is the derivative of angular position with respect to time. We start with the given angular position:
ϕ(t) = (0.326 rad/s²)t²
To find the angular speed, we differentiate ϕ with respect to time:
ω(t) = dϕ/dt = 2 * (0.326 rad/s²)t = 0.652t rad/s
Now, we can calculate the angular speed at t = 5.60 s:
ω(5.60) = 0.652 * 5.60 = 3.6512 rad/s
Tangential Speed Determination
Tangential speed (v) is related to angular speed and radius by the formula:
v = ω * r
Substituting the values we have:
v = 3.6512 rad/s * 10.4 m = 38.03 m/s
Calculating Tangential Acceleration
Tangential acceleration (a_t) is related to the angular acceleration (α) and the radius (r) by the formula:
a_t = α * r
To find angular acceleration, we differentiate angular speed:
α = dω/dt = 0.652 rad/s * 1 = 0.652 rad/s²
Now we can find the tangential acceleration:
a_t = 0.652 rad/s² * 10.4 m = 6.78 m/s²
Finding Radial Acceleration
Radial acceleration (a_r), also known as centripetal acceleration, can be calculated using the formula:
a_r = ω² * r
Substituting our values:
a_r = (3.6512 rad/s)² * 10.4 m = 138.13 m/s²
Summary of Results
- Angular Speed: 3.6512 rad/s
- Tangential Speed: 38.03 m/s
- Tangential Acceleration: 6.78 m/s²
- Radial Acceleration: 138.13 m/s²
In summary, by analyzing the motion of the centrifuge and applying the relevant equations of rotational dynamics, we can determine the various speeds and accelerations experienced by the astronaut during the test. Understanding these concepts is crucial in fields like astronaut training and engineering design for spacecraft and centrifuges.
