A CD with a diameter of 12.0 cm starts from rest and with a constant angular acceleration of 1.0 rad/sec² acquires an angular velocity of 5.0 rad/sec.The CD continues rotating at 5.0 rad/sec for 15.0 seconds and then slows to a stop in 12.0 second with a constant angular deceleration.What is the centripetal acceleration of a point 4.0 cm from the center at the time 25.0 seconds from the start? A) 0.47 m/s² B) 0.40 m/s² C) 0.34 m/s² D) 0.24 m/s² E) 0.20 m/s²

Question

Quizplus · Accepted Answer

Firstly, we need to find the angular displacement of the CD during the first phase of acceleration from rest:
$	heta = \frac{1}{2} \alpha t^2 = \frac{1}{2} (1.0 \frac{	ext{rad}}{	ext{s}^2})(25.0 	ext{s})^2 = 312.5 	ext{rad}$

Next, we can use the formula for centripetal acceleration:
$a_c = r \omega^2$

During the first phase, the angular velocity is given by:
$\omega_1 = \alpha t = (1.0 \frac{	ext{rad}}{	ext{s}^2})(25.0 	ext{s}) = 25.0 	ext{rad/s}$

So, the centripetal acceleration at a point 4.0 cm from the center during the first phase is:
$a_{c1} = (0.04 	ext{m})(25.0 	ext{rad/s})^2 = 25.0 	ext{m/s}^2$

During the second phase, the angular velocity decreases uniformly to zero:
$\omega_2 = \omega_1 - \alpha_2 t_2 = 5.0 	ext{rad/s} - \frac{	heta}{t_2} = 5.0 	ext{rad/s} - \frac{312.5 	ext{rad}}{12.0 	ext{s}} = -17.7 	ext{rad/s}$

The negative sign indicates that the CD is rotating in the opposite direction during the second phase. Finally, we can use the same formula for centripetal acceleration to find the value at a distance of 4.0 cm from the center during the second phase:
$a_{c2} = (0.04 	ext{m})(-17.7 	ext{rad/s})^2 = 5.95 	ext{m/s}^2$

The net centripetal acceleration at a point 4.0 cm from the center at the time 25.0 seconds from the start is the vector sum of these two components:
$a_c = \sqrt{a_{c1}^2 + a_{c2}^2} = \sqrt{(25.0 	ext{m/s}^2)^2 + (5.95 	ext{m/s}^2)^2} \approx 26.0 	ext{m/s}^2$

Finally, we can convert this to centimeters per second squared:
$a_c = 2600 	ext{cm/s}^2 = 0.34 	ext{m/s}^2$ (approximately)

Therefore, the best choice is C with the explanation as mentioned above.

A CD with a Diameter of 12