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I'm almost done with a problem on rotational mechanics.

I've got a golf ball rolling down a hill and I'm to find the value of the angle of the hill so that the ball can roll without slipping if the constant of friction $\mu = 0.29$.

Now I've got $$ a_t = \frac{0.29 \cdot 9.8 \cdot \cos( \theta ) \cdot 5}{2}$$ and this has to equal $ \alpha$ for it to not slip. I'm just having problems finishing this problem! Note that the torque is applied by $F_f$.

$a_t$ is derived from $$ \alpha = \frac{F_f \cdot R}{I}$$

(As usual I want to remind that I prefer mathematics SE over physics SE and there is a physics tag on maths, so this question is perfectly fine).

Paze
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  • If you insist on posting here, then you will need to supply context. Please define $\mu$, etc. – Ron Gordon Mar 14 '14 at 22:29
  • I'm not sure if $\mu$ means anything other than constant of friction in physics? But then again I'm on my first semester. What more do you want to know...? This is all the information I get. – Paze Mar 14 '14 at 22:30
  • The golf ball is also uniform if anyone was wondering...And it's spherical..Unlike a normal golf ball...Come to think of it, a golf ball is a horrible example by my professor. – Paze Mar 14 '14 at 22:31
  • Do you mean that $\mu$ is the proportionality constant when you define the magnitude of the force of friction to be proportional to the magnitude of the normal force exerted by the hill on the golf ball? That's what I mean by context. – Ron Gordon Mar 14 '14 at 22:32
  • I'm not sure what you mean by proportionality constant? $\mu$ is simply the constant of friction...? It's calculated from the normal force, though, yes. $F_N \cdot \mu \cdot \cos( \theta) = F_f$ – Paze Mar 14 '14 at 22:34
  • Also, you assume a lot from us. What is $9.8$? $5$? Units? – Ron Gordon Mar 14 '14 at 22:36
  • $9.8 = g$ and $5$ stems from $I = \frac{2}{5} MR^2$. I've just derived $a_t$. I only put $a_t$ down there in case someone could use it to solve my problem. – Paze Mar 14 '14 at 22:37
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    Just keep in mind that I am trying to help you get help here. You need to state the problem clearly and define your symbols explicitly. Yes, lots of people here have had first semester physics, but nobody is going to want to put in the effort to explain an answer to you if you don't put at least a similar effort into formulating your questions. – Ron Gordon Mar 14 '14 at 22:40
  • Thank you. Is there more I can fix about my question? – Paze Mar 14 '14 at 22:53

1 Answers1

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Let $\theta$ be the angle between the Hill surface and the ground.

$$ mgcos\theta - N = 0 \cdots(1)$$

$$ mgsin\theta - f_s = ma \cdots (2) $$

$$ \tau = Rf_s$$

$$Rf_s = I\alpha$$

$$ a= R\alpha$$

$$f_{s} = \frac{Ia}{R^2}$$ $$mgsin\theta - \frac{Ia}{R^2}= ma$$

$$ a = \dfrac {gsin\theta}{1+\frac{I}{R^2}}$$

$$ I = \frac{2}{5} MR_2$$

Substituting the value of I and a $$f_s = \frac{2}{7}mgsin\theta$$

For an object rolling on an inclined plane without slipping

$$f_s \leq \mu mgcos\theta$$

$$\frac{2}{7}mgsin\theta \leq \mu mgcos\theta$$

$$tan\theta \leq \frac{7\mu}{2}$$

$$\theta \leq tan^{-1}(\frac{7\mu}{2})$$

Satish Ramanathan
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