Finding centripetal acceleration of a CD

[bjudia]

1. 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 10.0 seconds from the start?
2. I know constant angular acceleration equations ωf=ωi+αt
Also centripetal acceleration= ω^2r

I really need some help with this, my professor did not explain this at all. I know radius 0.06m. ωi=5.0rad/sec. The answer to the solution is 1.0 m/s^2 but I'm not sure how to get there.

 

[Nessdude14]

You're given the angular velocity (5 rad/sec.) and the radius that the point revolves around (4 cm or .04 m). You just need to plug the values into the centripetal acceleration equation (a=ω²r)

 

[PhanthomJay]

First correct your typo for the radius at the point in question, which is 0.04 m (.06 m is the radius of the disc).

Calculate the angular speed after 10 seconds, then plug it into the appropriate equation for centripetal acceleration. First you must answer: how long does it take reach its speed of 5 rad/sec? You've listed the relevant equations...now please show an attempt.

 

[bjudia]

That makes total sense! I've been blown away for like 2 hours. lol thank you very much

 

[Nickie]

I can provide a response to the content by first understanding the given information and then using relevant equations to solve the problem.

Firstly, we are given a CD with a diameter of 12.0 cm, which means the radius is half of that, or 6.0 cm (0.06 m). The CD starts from rest, meaning its initial angular velocity (ωi) is 0. It then acquires an angular velocity of 5.0 rad/sec after a constant angular acceleration of 1.0 rad/sec^2.

Using the equation ωf=ωi+αt, we can calculate the final angular velocity (ωf) after 10.0 seconds from the start:

ωf=ωi+αt
ωf=0+1.0 rad/sec^2 * 10.0 sec
ωf=10.0 rad/sec

Now, we are asked to find the centripetal acceleration at a point 4.0 cm from the center after 10.0 seconds. We can use the equation for centripetal acceleration, ac= ω^2r, to solve for this.

ac= ω^2r
ac= (10.0 rad/sec)^2 * 0.04 m
ac= 1.0 m/s^2

This matches the given answer of 1.0 m/s^2. Therefore, the centripetal acceleration at a point 4.0 cm from the center after 10.0 seconds is 1.0 m/s^2.

Additionally, we are told that the CD continues rotating at 5.0 rad/sec for 15.0 seconds and then slows to a stop in 12.0 seconds with a constant angular deceleration. We can use the same equation to find the angular deceleration (α) and then use it to find the time it takes for the CD to come to a stop.

ωf=ωi+αt
0=5.0 rad/sec+α * 12.0 sec
α= -0.417 rad/sec^2

Now, we can use this angular deceleration to find the time it takes for the CD to come to a stop:

ωf=ωi+αt
0=5.0 rad/sec+(-0.417 rad/sec^2) * t
t= 12.0