A cylinder rolls without slipping- find the angle

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SUMMARY

A cylinder with radius R and mass M rolls down an incline plane at an angle θ without slipping, with a coefficient of friction μ. The moment of inertia for the cylinder is given by I=(MR²)/2. The conservation of energy equation leads to the relationship gh=3/4v², while the forces acting on the cylinder include gravitational force components and normal force. The maximum angle for rolling without slipping is determined by the balance of forces and the maximum static friction.

PREREQUISITES
  • Understanding of rotational dynamics and moment of inertia
  • Knowledge of energy conservation principles in physics
  • Familiarity with forces acting on objects on inclined planes
  • Basic algebra for solving equations
NEXT STEPS
  • Study the derivation of the conservation of energy in rolling motion
  • Learn about static friction and its role in rolling without slipping
  • Explore the equations of motion for objects on inclined planes
  • Investigate the effects of varying the angle of inclination on rolling objects
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Students studying physics, particularly those focusing on mechanics, as well as educators looking for examples of rolling motion and friction in real-world applications.

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Homework Statement



A cylinder with radius R and mass M rolls without slipping down an incline plane with angle\theta. The coeff. of friction is \mu. Find the maximum value for the angle for the cylinder to roll without slipping.

Homework Equations



The moment of inertia for a cylinder is I=(MR^2)/2.
w=v/r

The Attempt at a Solution



If the cylinder rolls without slipping, energy must be conserved. So Mgh=1/2Mv^2+1/2Iw^2. If you plug in the equations from above and divide all terms by M, gh=3/4v^2.
So we need to find v.
The equation for the x-component of force (where x is along the plane of the incline), F=mg(cos\theta-\mu)=ma
and the y-component
F=0=N-mgsin\theta or N=mgsin\theta

I have no idea where to go from here and I'm not quite sure I did all of this right. Thank you so much for any help!
 
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You should think when you will have the minimum value of angle.When you have the maximum of static friction.Also i think you have a mistake.In the direction of the slope the forces are Wy=mgsinθ and T<=μN
 
Also N=mgcosθ not mgsinθ
 

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