Well, it seems like you're facing a fan-tastic physics problem! Let's dive into it, shall we?
First, let's figure out the current angular position of the fan blades. We can use the formula:
θ = θ0 + ω0t + (1/2)αt²
where:
θ is the angular position,
θ0 is the initial angular position (which we assume to be 0),
ω0 is the initial angular velocity (1.36 rad/s),
α is the angular acceleration (1.82 rad/s²),
and t is the elapsed time (0.475 s).
Plugging in the values, we get:
θ = 0 + (1.36 rad/s)(0.475 s) + (1/2)(1.82 rad/s²)(0.475 s)²
Calculating this, we find that the current angular position is approximately 0.745 radians.
Next, let's determine the current tangential speed of a point on the edge of the fan blade. We can use the formula:
v = rω
where:
v is the tangential speed,
r is the radius of the fan blade (0.393 m),
and ω is the angular velocity (1.36 rad/s).
Substituting in the values, we get:
v = (0.393 m)(1.36 rad/s)
Calculating this, we find that the current tangential speed is approximately 0.534 m/s.
Lastly, let's find the current centripetal acceleration of a point on the edge of the fan blade. We can use the formula:
a = rα
where:
a is the centripetal acceleration,
r is the radius of the fan blade (0.393 m),
and α is the angular acceleration (1.82 rad/s²).
Substituting in the values, we get:
a = (0.393 m)(1.82 rad/s²)
Calculating this, we find that the current centripetal acceleration is approximately 0.716 m/s².
So, after 0.475 seconds since the switch was reset, the following values are approximately:
- Angular position: 0.745 radians
- Tangential speed: 0.534 m/s
- Centripetal acceleration: 0.716 m/s²
Keep those fan-tastic physics problems coming!