Question

A high school physics instructor catches one of his students chewing gum in class. He decides to discipline the student by asking that he stick the gum to a fan and calculate how fast the fan is moving when the gum gets thrown off. The label says that the diameter of the fan is d= 21 cm, and at full speed it turns at a rate off-35 rev/s, and that the fan is guaranteed to accelerate uniformly. The fan takes 13 s to go from rest to full speed. Randomized Variables d=21 cm f 35 rev/s t=13 s ?? 14%

Part (a) Calculate the maximum the angular velocity of the fan ohax, in radians per second. ? 14%
Part (b) Surprisingly, the gum seems to remain stuck to the fan at this speed. Calculate the angular acceleration of the gum ? per square second, as the fan is speeding up. in radians 1 4%
Part (c) Calculate the tangential component of the acceleration of the gum atan in meters per square second, as the fan is speeding up. ? 14%
Part (d) what is the magnitude of the centripetal acceleration of the gum a d in meters per square second when the fan reaches full speed? 14%
Part (e) what is the direction of the centripetal acceleration of the gum, as the fan is turning at top speed? ? ? 14%
Part (f) Calculate the tangential component of the acceleration of the gum a in meters per square second when the fan is at full speed ? 14%
Part (g) Soon after reaching this speed, the gum becomes un stuck from the fan blade. Determine the linear speed of the gum v, in meters per second, immediately after it leaves the fan

Answer 1

Answer:

a) $\omega \approx 219.911\,\frac{rad}{s}$, b) $\alpha = 16.916\,\frac{rad}{s^{2}}$, c) $a_{t} = 1.776\,\frac{m}{s^{2}}$, d) $a_{n} = 5077.889\,\frac{m}{s^{2}}$, e) The direction of the centripetal acceleration experimented by the gum goes to the center of rotation, f) Zero, g) $v = 23.091\,\frac{m}{s}$.

Explanation:

a) The maximum angular velocity of the fan is:

$\omega = (35\,\frac{rev}{s} )\cdot (\frac{2\pi\,rad}{1\,rev} )$

$\omega \approx 219.911\,\frac{rad}{s}$

b) The angular acceleration of the fan is:

$\alpha = \frac{\omega-\omega_{o}}{t}$

$\alpha = \frac{219.911\,\frac{rad}{s}-0\,\frac{rad}{s}}{13\,s}$

$\alpha = 16.916\,\frac{rad}{s^{2}}$

c) The magnitude of the tangential aceleration is:

$a_{t} = (16.916\,\frac{rad}{s^{2}} )\cdot (0.105\,m)$

$a_{t} = 1.776\,\frac{m}{s^{2}}$

d) The magnitude of the centripetal acceleration is:

$a_{n} = (219.911\,\frac{rad}{s} )^{2}\cdot (0.105\,m)$

$a_{n} = 5077.889\,\frac{m}{s^{2}}$

e) The direction of the centripetal acceleration experimented by the gum goes to the center of rotation.

f) When fan is at full speed, it rotates at constant rate and, hence, there is no angular acceleration. Besides, the tangential acceleration experimented by the gum is zero.

g) The linear speed of the gum is:

$v = (219.911\,\frac{rad}{s} )\cdot (0.105\,m)$

$v = 23.091\,\frac{m}{s}$