Question

An electric ceiling fan is rotating about a fixed axis with an initial angular velocity magnitude of 0.270 rev/s . The magnitude of the angular acceleration is 0.895 rev/s^2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.790 m .

1. Compute the fan's angular velocity magnitude after time 0.194s has passed. (rev/s)
2. Through how many revolutions has the blade turned in the time interval 0.194s from Part A? (rev)
3. What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.194s ? (m/s)
4. What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.194s ? (m/s^2)

Answer 1

Answer:

Given information is summarized as below:

$\omega _{o}=0.270rev/sec=1.69 rad/sec\\\alpha =0.895rev/sec=5.62rad/sec\\d=0.790m$

1)Using first equation of kinematics for angular motion we have

$\omega _{f}=\omega _{o}+\alpha t\\\\\therefore \omega _{f}=1.69+5.62\times 0.194\\\omega _{f}=2.780rad/sec$

2) Using second equation of kinematics for angular motion we have

$\theta =\omega _{o}t+\frac{1}{2}\alpha t^{2}\\\\\theta =1.69\times 0.194+05\times 5.62\times0.194^{2}\\\\\theta = 0.4336rad$

3) The tangential speed is given as

$v_{t}=\omega _{f}\times r\\\\v_{t}=2.78\times\frac{0.790}{2}=1.098m/s$

4)

The resultant acceleration is given by

$a_{res}=\sqrt{(a_{tangential})^{2}+(a_{radial})^{2}}\\\\a_{res}=\sqrt{(\alpha r)^{2}+(\omega _{f}^{2}r)^{2}}\\\\a_{res}=3.77m/s^{2}$

Answer 2

The centripetal force is the force that keeps an object moving along a circular path.

Using the formula;

ωf = ωi - αt

Where;

ωf = final angular velocity

ωi = Initial angular velocity

α = angular acceleration

t = time taken

Substituting values;

ωf = 0.270 rev/s +  (0.895 rev/s^2 × 0.194s)

ωf = 0.443 rev/s

Given that;

ω = θ/t

0.443 rev/s = θ/ 0.194s

θ = 0.443 rev/s  × 0.194s

θ =  0.086 rev

The tangential speed is obtained from

v = rω

Since 1 rev/s = 2π rad/s

0.443 rev/s = 0.443 rev/s × 2π rad/s/1 rev/s

= 2.78 rad/s

v = (0.790 m/2) × 2.78 rad/s

v = 1.098 m/s

ac = v^2/r

ac= ( 1.098 m/s)^2/0.395

ac = 3.05 m/s^2

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