The blades of a ceiling fan have a radius of 0.340m and are rotating about a fixed axis with an intial angular velocity of +1.25rad/s2. When the switch on the fan is turned to a higher speed, the blades acire an angular acceleration of +2.55rad/s2. After 0.480s have elapsed since the switch was reset, what are
A) the magnitude of the total acceleration (in m/s2)of a poit on the tip of a blade, and
B) the angle theta between the total acceleration, a, and the centripetal accleration, ac?
To solve this problem, we need to understand the different types of acceleration involved. We have the angular acceleration, which is given as +2.55 rad/s^2, and we need to find the total acceleration and the angle between the total acceleration and the centripetal acceleration.
To find the answers, we will use the following formulas:
1. Angular acceleration (α) = Δω / Δt
2. Angular velocity (ω) = ω0 + αt
3. Tangential acceleration (at) = r * α
4. Centripetal acceleration (ac) = r * ω^2
5. Total acceleration (a) = sqrt(at^2 + ac^2)
Let's solve it step by step:
Step 1: Finding Angular Velocity (ω)
Using formula (2), we can find the angular velocity (ω) when 0.480 seconds have elapsed since the switch was reset:
ω = ω0 + α * t
= 1.25 rad/s^2 * 0.480 s
= 0.6 rad/s
Step 2: Finding Centripetal Acceleration (ac)
Using formula (4), we can calculate the centripetal acceleration:
ac = r * ω^2
= 0.340 m * (0.6 rad/s)^2
= 0.366 m/s^2
Step 3: Finding Tangential Acceleration (at)
Using formula (3), we can calculate the tangential acceleration:
at = r * α
= 0.340 m * 2.55 rad/s^2
= 0.867 m/s^2
Step 4: Finding Total Acceleration (a)
Using formula (5), we can calculate the total acceleration:
a = sqrt(at^2 + ac^2)
= sqrt((0.867 m/s^2)^2 + (0.366 m/s^2)^2)
= sqrt(0.751 + 0.134)
= sqrt(0.885)
≈ 0.941 m/s^2
Therefore, the magnitude of the total acceleration is approximately 0.941 m/s^2.
Step 5: Finding the Angle Theta (θ)
To find the angle between the total acceleration (a) and the centripetal acceleration (ac), we can use trigonometry:
tan(θ) = at / ac
θ = arctan(at / ac)
= arctan(0.867 m/s^2 / 0.366 m/s^2)
≈ 66.25 degrees
Therefore, the angle (θ) between the total acceleration (a) and the centripetal acceleration (ac) is approximately 66.25 degrees.
In summary:
A) The magnitude of the total acceleration is approximately 0.941 m/s^2.
B) The angle theta (θ) between the total acceleration (a) and the centripetal acceleration (ac) is approximately 66.25 degrees.