Question

A uniform sphere of mass M and radius r rolls without slipping inside a track with the cross-section of a semi-circle of radius R (R>r) The sphere's motion is confined to a vertical plane. In the figure below, point B becomes point C when the sphere is at the angular position $\Theta = 0^\circ$. Let $\phi$ be the angular displacement of the sphere about its center. (a) Write down constraints for the sphere's motion. (b) Derive the equations of motion from the Lagrangian. (c) Determine and interpret the constraint forces. (d) Determine the period oscillations of the sphere about the equilibrium position ($\Theta = 0^\circ$ is the equilibrium position), assuming small angular displacements. (e) Determine the period oscillations of the sphere about the equilibrium position if no force of friction exists between the sphere and the track?

          A uniform sphere of mass M and radius r rolls without slipping inside a track with the
cross-section of a semi-circle of radius R (R>r) The sphere's motion is confined to a vertical
plane. In the figure below, point B becomes point C when the sphere is at the angular
position $\Theta = 0^\circ$. Let $\phi$ be the angular displacement of the sphere about its center.
(a) Write down constraints for the sphere's motion.
(b) Derive the equations of motion from the Lagrangian.
(c) Determine and interpret the constraint forces.
(d) Determine the period oscillations of the sphere about the equilibrium position ($\Theta = 0^\circ$
is the equilibrium position), assuming small angular displacements.
(e) Determine the period oscillations of the sphere about the equilibrium position if no
force of friction exists between the sphere and the track?

A uniform sphere of mass M and radius r rolls without slipping inside a track with the
cross-section of a semi-circle of radius R (R>r) The sphere's motion is confined to a vertical
plane. In the figure below, point B becomes point C when the sphere is at the angular
position Θ = 0^∘. Let ϕ be the angular displacement of the sphere about its center.
(a) Write down constraints for the sphere's motion.
(b) Derive the equations of motion from the Lagrangian.
(c) Determine and interpret the constraint forces.
(d) Determine the period oscillations of the sphere about the equilibrium position (Θ = 0^∘
is the equilibrium position), assuming small angular displacements.
(e) Determine the period oscillations of the sphere about the equilibrium position if no
force of friction exists between the sphere and the track?

Added by Nancy H.

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