# Finding the tangential component of acceleration

1. Jan 31, 2010

### jumbogala

EDIT: I meant radial in the title.
1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. The attempt at a solution
My book says that to use the formula ar= 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:
wfinal = winitial + $$\alpha$$(t)
wf = 6.5 rad/s $$\hat{k}$$

Then use that first formula:
ar = (169 rad/sm)$$\hat{k}$$

Are those units correct? Really the formula for ar = 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?

Last edited: Jan 31, 2010
2. Jan 31, 2010

### tiny-tim

Hi jumbogala!

(have an alpha: α and an omega: ω )

you only have 1.5 s of acceleration at 2s.

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.
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).

3. Jan 31, 2010

### 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).

4. Jan 31, 2010

### tiny-tim

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