Respuesta :
Answer:
The speed of the tip of a blade 9.6 s after the fan is turned off is 1.23 m/s
Explanation:
Given;
diameter of ceiling fan blade, d = 67-cm
initial angular velocity, ωi = 54 rpm
time required for the blade to stop, t = 27.3 s
Apply kinematic equation;
ωf = ωi + αt
where;
ωf is the final angular velocity = 0
ωi is the initial angular acceleration = 54 rev/min x 2π/60s = 5.656 rad/s
α is the angular acceleration of the fan
ωf = ωi + αt
0 = 5.656 +α(27.3)
α(27.3) = -5.656
α = -5.656/27.3
α = -0.207 rad/s²
Then, angular acceleration of the fan after 9.6 seconds is calculated as;
ω' = ωi + αt
ω' = 5.656 + (-0.207 x 9.6)
ω' = 5.656 - 1.987
ω' = 3.669 rad/s
Finally, the speed of the tip of a blade 9.6 s after the fan is turned off;
V = ωr
where
r is the radius of the blade = ¹/₂ x d = ¹/₂ x 67 = 33.5 cm = 0.335 m
V = ωr = (3.669 rad/s) x (0.335 m)
V = 1.23 m/s
Answer:
1.228 m/s
Explanation:
In rotational kinematics, we have the following equations;
ω_f = ω_i + αt - - - - (1)
Where;
ω_f is final angular velocity
ω_i is initial angular velocity
α is angular acceleration
t is time taken
Also; instantaneous velocity is given by;
v = rω - - - - - (eq2)
Where r is the distance from the angle of rotation
We are given;
ω_i = 54 rpm = 54 x 0.10472 rad/s =
5.655 rad/s
Time required for it to stop running after turning it; t = 27.3 s
Diameter(D) = 67cm = 0.67m
Thus, let's find the angular acceleration by plugging in the relevant values into eq 1
Since it comes to rest after 27.3s,thus ω_f = 0
0 = 5.655 + 27.3α
-27.3α = 5.655
α = -5.655/27.3 = -0.2071 rad/s²
Now, we want to find the speed after 9.6s
Thus,
ω_f = 5.655 + 9.6(-0.2071)
ω_f = 5.655 - 1.988 = 3.667 rad/s
Now, since the diameter is 0.67m,then the distance between the tip and the angle of rotation is;
r = d/2 = 0.67/2 = 0.335m
So, for the final velocity of the tip of a blade 9.6 s after the fan is turned off; let's plug in the relevant values into eq 2.
v = rω = 0.335 x 3.667 = 1.228 m/s
