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A sphere lies on a frictionless, horizontal surface. A horizontal force is applied to the sphere (but above or below its center of mass).

  • Will the sphere roll or slide? (I think it will roll based on this answer)
  • If it does roll, then will the acceleration be different compared to if the force is applied to the center of mass?
Starlight
  • 347

1 Answers1

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lets write the equations

$$ M\dot v=F\\ I_C\dot\omega=F\,\rho$$

where $~\rho~$ the distance from the center of mass to the force $~F~$

the velocity at the contact point $~v_c~$ is

$$ v_c=\omega\,r-v \tag 1$$

thus,

  • $~v_c=0~$ sphere is rolling
  • $~v_c > 0~$ sphere is sliding

from the EOM's

$$ v=\frac{F}{M}\,t\\ \omega=\frac{F\,\rho}{I_C}\,t$$

from here with $~I_C=\frac 25 M\,r^2~$ you obtain

$$ v_c=\frac 52\,\frac \rho r-1,\quad \rho < r$$

thus if $~\rho=\frac 25 r~$ the sphere is rolling and if $~\frac 25 r < \rho < 1~$ the sphere is sliding

Eli
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