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Sphere rolling down an incline

  • Thread starter dk214
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  • #1
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Homework Statement


A hollow spherical shell with mass 2.50 kg rolls without slipping down a slope that makes an angle of 32.0 degrees with the horizontal.
a. Find the magnitude of the acceleration [tex] a_c[/tex] of the center of mass of the spherical shell.
b. Find the magnitude of the frictional force acting on the spherical shell.
c. Find the minimum coefficient of friction [tex] \mu [/tex] needed to prevent the spherical shell from slipping as it rolls down the slope.


Homework Equations


For part a.
Since its pure roll, [tex] a_c = \alpha * R
\alpha = a_c/R [/tex]
[tex] \tau = R*Friction = I (moment-of-inertia) * \alpha [/tex]
[tex]Friction = (I*\alpha)/R = (I*a_c)/R^2 [/tex]
[tex]Ma_c = Mgsin(\theta)-Friction[/tex]
[tex]Ma_c = Mgsin(\theta)-Ia_c/R^2[/tex]
[tex]a_c = (MR^2*g*sin(\theta))/(MR^2+I)[/tex]


The Attempt at a Solution


I for sphere =[tex] 2/3 MR^2[/tex]
so, [tex]a_c = (MR^2*g*sin(\theta))/(MR^2+2/3*MR^2)[/tex]
MR^2 cancels..
[tex]a_c = 3/5*g*sin(\theta)[/tex]
for a_c i got [tex]a_c = 3.12m/s^2[/tex] i think im right unless i made a mathematical error some where.
and substituting a_c, in [tex]Ma_c = Mgsin(\theta)-Friction[/tex]
i got Friction = 5.19 N.
And c,
this where I'm kind of stuck. I'm assuming since they are asking for minimum [tex]\mu[/tex] Friction is 0 in [tex] Ma_c = Mgsin(\theta)-Friction [\tex]
[tex]a_c = gsin(\theta) [/tex].
[tex]Friction = (I*\alpha)/R = (I*a_c)/R^2 [/tex], and
[tex] Friction = \mu*mg*sin(\theta) [/tex]
[tex] /mu= ((I*a_c)/R^2)/mg*sin(\theta)
idk if I'm right in assuming Friction is 0 in one part and not in other.. Any hints/guides and help would greatly be appreciated.
 
Last edited:

Answers and Replies

  • #2
alphysicist
Homework Helper
2,238
1
Hi dk214,

The Attempt at a Solution


I for sphere =[tex] 2/3 MR^2[/tex]
so, [tex]a_c = (MR^2*g*sin(\theta))/(MR^2+2/3*MR^2)[/tex]
MR^2 cancels..
[tex]a_c = 3/5*g*sin(\theta)[/tex]
for a_c i got [tex]a_c = 3.12m/s^2[/tex] i think im right unless i made a mathematical error some where.
and substituting a_c, in [tex]Ma_c = Mgsin(\theta)-Friction[/tex]
i got Friction = 5.19 N.
And c,
this where I'm kind of stuck. I'm assuming since they are asking for minimum [tex]\mu[/tex] Friction is 0 in
I don't think this is right; you've already found the force of friction. Now they want the minimum [itex]\mu[/itex] that can supply that force; in other words they want the coefficient for which that frictional force is a maximum. What does that give?
 
  • #3
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I dont know if I'm understanding the question right. Are they just asking for the [tex] \mu [/tex] for the friction I found.?
which would just be Friction/Normal
[tex] \mu = 5.12/Mgcos(\theta) [/tex]
[tex] \mu = .246[/tex]
 

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