Finding the tangential component of acceleration

[jumbogala]

EDIT: I meant radial in the title.

Homework Statement

A ball is going around in a circle of radius 4 m.

It goes with a constant angular velocity of (13 rad/s)$$\hat{k}$$ for 0.5 s. After that, it takes 4 s to come to a complete stop.

Find the radial component of the ball's acceleration at 2 s.

Homework Equations

The Attempt at a Solution

My book says that to use the formula a_r= w²r. However, w is changing, so I don't see how I can use that!

The only thing I can think of is to find the angular acceleration:
$$\alpha$$ = w₀ + $$\alpha$$₀(t)
0 = (13 rad/s) + $$\alpha$$₀(4 s). Solving for $$\alpha$$ gives -3.25 rad/s²$$\hat{k}$$

Then I use another formula to find the angular velocity at 2 s:
w_final = w_initial + $$\alpha$$(t)
w_f = (13 rad/s) + (-3.25 rad/s²)(2 s)
w_f = 6.5 rad/s $$\hat{k}$$

Then use that first formula:
a_r = (6.5 rad/s)²(4 m)
a_r = (169 rad/sm)$$\hat{k}$$

Are those units correct? Really the formula for a_r = dVt / dt, but is what I did ok?

Also, as an aside, the TANGENTIAL part of the angular acceleration would stay the same all the time, right? If I calculated it at 1 s, 2s, ... 4.3 s, it would not change?

 

[tiny-tim]

Hi jumbogala!

(have an alpha: α and an omega: ω )

… It goes with a constant angular velocity of (13 rad/s)$$\hat{k}$$ for 0.5 s. After that, it takes 4 s to come to a complete stop.

Find the radial component of the ball's acceleration at 2 s.

Your calculations are fine, except that you've misread the question …

you only have 1.5 s of acceleration at 2s.

Are those units correct? Really the formula for a_r = dVt / dt, but is what I did ok?

You're right to be worried … the units in the formula v = ωr are cm/s = rad/s times cm … and in the formula a = ω²r are cm/s² = rad²/s² times cm … the radians are dimensionless, and they just drop out.

Also, as an aside, the TANGENTIAL part of the angular acceleration would stay the same all the time, right? If I calculated it at 1 s, 2s, ... 4.3 s, it would not change?

Not following this.

"tangential part of the angular acceleration" makes no sense.

Do you mean the tangential part of the ordinary acceleration (ie, the tangential acceleration)?

If so, then yes, you're correct … for fixed radius, that's simply dv/dt, the derivative of the speed (= r dω/dt = rα).

 

[jumbogala]

Thank you!

I'm confused about those units still, though. Why are we using cm, if the radius is given in m? Is that just a convention?

Also, the rad drops out for a, but if I just want to write ω, can I still write rad/s? (Instead of 1/s).

 

[tiny-tim]

I'm confused about those units still, though. Why are we using cm, if the radius is given in m? Is that just a convention?

oh, I made a mistake … I thought the question used cm.

Also, the rad drops out for a, but if I just want to write ω, can I still write rad/s? (Instead of 1/s).

Yes, ω is rad/s.