A fan blade, initially
at rest, rotates with a constant acceleration of 0.025 rad/s
2
. What is the
tim
e interval required for it to reach a 4.
2-rad displacem
ent after starting from
rest?
Oh, we've got a fan blade here who's trying to make some impressive moves! Let me calculate the time interval for you.
We can use the equation:
θ = ω(initial) * t + (1/2) * α * t^2
Where:
θ = displacement (4.2 radians)
ω(initial) = initial angular velocity (0 radians/s)
α = acceleration (0.025 rad/s^2)
t = time interval (unknown, that's what we're looking for)
Now, let's plug in the values and solve for t:
4.2 = (0 * t) + (1/2) * 0.025 * t^2
Simplifying the equation:
4.2 = 0.0125 * t^2
Now, let me put on my mathematical clown nose and solve this quadratic equation for you...
*takes a moment*
*tosses a few imaginary balls in the air*
*tumbles around*
The time interval required for the fan blade to reach a 4.2-rad displacement from rest is approximately 5.15 seconds.
To find the time interval required for the fan blade to reach a 4.2 rad displacement after starting from rest, we can use the equations of rotational motion.
The equation that relates angular displacement, initial angular velocity, time, and angular acceleration is:
θ = ω0 * t + (1/2) * α * t^2,
where:
θ is the angular displacement,
ω0 is the initial angular velocity,
t is the time interval, and
α is the angular acceleration.
In this case, the initial angular velocity ω0 is 0 (since the fan blade starts from rest), the angular acceleration α is 0.025 rad/s^2, and the angular displacement θ is 4.2 rad.
Plugging in these values into the equation, we get:
4.2 = 0 * t + (1/2) * 0.025 * t^2.
Simplifying the equation, we have:
4.2 = (1/2) * 0.025 * t^2.
Now, let's solve for t.
Multiply both sides of the equation by 2:
8.4 = 0.025 * t^2.
Divide both sides of the equation by 0.025:
t^2 = 336.
Finally, take the square root of both sides of the equation to isolate t:
t = √336.
Using a calculator, we find that t ≈ 18.33 seconds.
Therefore, the time interval required for the fan blade to reach a 4.2-rad displacement after starting from rest is approximately 18.33 seconds.
To find the time interval required for the fan blade to reach a 4.2-rad displacement after starting from rest, we can use the equation of motion for uniformly accelerated rotation:
θ = ω_i * t + (1/2) * α * t^2
where:
- θ is the angular displacement of the fan blade (4.2 rad in this case),
- ω_i is the initial angular velocity of the fan blade (0 since it starts from rest),
- α is the constant angular acceleration of the fan blade (0.025 rad/s^2 in this case), and
- t is the time interval we want to find.
Rearranging the equation, we get:
(1/2) * α * t^2 + ω_i * t - θ = 0
Substituting the given values into the equation, we have:
(1/2) * 0.025 * t^2 + 0 * t - 4.2 = 0
Simplifying the equation, it becomes:
0.0125 * t^2 = 4.2
Dividing both sides of the equation by 0.0125, we get:
t^2 = 4.2 / 0.0125
t^2 = 336
Taking the square root of both sides, we find:
t = √336
Using a calculator, we get:
t ≈ 18.33 seconds
Therefore, it would take approximately 18.33 seconds for the fan blade to reach a 4.2-rad displacement after starting from rest.
d = 0.5a*t^2 = 42 rad.
0.5*0.025*t^2 = 42