The blades of a ceiling fan have a radius of 0.393 m and are rotating about a fixed axis with an angular velocity of +1.36 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of 1.82 rad/s2. After 0.475 s have elapsed since the switch was reset, what are the following?
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!
To find the following values, we need to use the formulas of rotational motion:
1. Angular displacement (θ):
θ = ωi * t + (1/2) * α * t^2
where:
θ is the angular displacement,
ωi is the initial angular velocity,
t is the time, and
α is the angular acceleration.
Plugging in the given values, we get:
θ = (1.36 rad/s) * (0.475 s) + (1/2) * (1.82 rad/s^2) * (0.475 s)^2
Evaluating this equation, we find θ.
2. Final angular velocity (ωf):
ωf = ωi + α * t
Plugging in the given values, we get:
ωf = (1.36 rad/s) + (1.82 rad/s^2) * (0.475 s)
Evaluating this equation, we find ωf.
3. Total angle turned (Δθ):
Δθ = ωi * t + (1/2) * α * t^2
where Δθ is the total angle turned.
Plugging in the given values, we get:
Δθ = (1.36 rad/s) * (0.475 s) + (1/2) * (1.82 rad/s^2) * (0.475 s)^2
Evaluating this equation, we find Δθ.
4. Total time taken to stop:
To find the total time taken to stop, we need to find the time when the blades will stop rotating. At this point, the final angular velocity (ωf) will be zero.
ωf = 0
Using this equation, we can determine the time it takes for the blades to stop.
Please note that these calculations assume that the angular acceleration is constant throughout the motion.