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Problem 3. Hoop rolling down an inclined plane (6 p = 15%) A ring of mass $m$ and radius $R$ rolls down an inclined plane without slipping (see Figure). Let us denote its moment of inertia simply $I$. The inclined plane (mass $M$) has an opening angle $\alpha$ and it slides itself frictionlessly along a horizontal surface (instantaneous position $X$). (a) Figure out the appropriate generalized coordinates and write out the Lagrangian function. [3p] Hint: Start from the position coordinates for the center of the ring. Use the slipping constraint the same way as we are used to do with holonomic constraints (note: no forces requested). (b) Solve the equations of motion and resulting accelerations for the ring and inclined plane. [3p]

          Problem 3. Hoop rolling down an inclined plane (6 p = 15%)
A ring of mass $m$ and radius $R$ rolls down an
inclined plane without slipping (see Figure). Let us
denote its moment of inertia simply $I$. The
inclined plane (mass $M$) has an opening angle $\alpha$
and it slides itself frictionlessly along a horizontal
surface (instantaneous position $X$).
(a) Figure out the appropriate generalized
coordinates and write out the Lagrangian
function. [3p]
Hint: Start from the position coordinates for the center of the ring. Use the slipping constraint
the same way as we are used to do with holonomic constraints (note: no forces requested).
(b) Solve the equations of motion and resulting accelerations for the ring and inclined plane. [3p]

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Problem 3. Hoop rolling down an inclined plane (6 p = 15%)
A ring of mass m and radius R rolls down an
inclined plane without slipping (see Figure). Let us
denote its moment of inertia simply I. The
inclined plane (mass M) has an opening angle α
and it slides itself frictionlessly along a horizontal
surface (instantaneous position X).
(a) Figure out the appropriate generalized
coordinates and write out the Lagrangian
function. [3p]
Hint: Start from the position coordinates for the center of the ring. Use the slipping constraint
the same way as we are used to do with holonomic constraints (note: no forces requested).
(b) Solve the equations of motion and resulting accelerations for the ring and inclined plane. [3p]

Added by Nicholas M.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

03/30/2023

Step 1

Step 1: Generalized coordinates We can use two generalized coordinates to describe the system: the position of the center of the ring along the inclined plane, denoted by x, and the angle of rotation of the ring about its center, denoted by θ. Show more…

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Both the plane and the ring is moving. Please find the appropriate generalized coordinates and explain the geometry behind how you came to the answer. After that write the lagrangian and find the equations of motion for both the ring and inclined plane. Problem 3. Hoop rolling down an inclined plane (6 p = 15%) A ring of mass m and radius R rolls down an m,1 inclined plane without slipping (see Figure). Let us denote its moment of inertia simply I. The inclined plane (mass M) has an opening angle a and it slides itself frictionlessly along a horizontal surface (instantaneous position X). M (a) Figure out the appropriate generalized coordinates and write out the Lagrangian function. [3p] X Hint: Start from the position coordinates for the center of the ring. Use the slipping constraint the same way as we are used to do with holonomic constraints (note: no forces requested) (b) Solve the equations of motion and resulting accelerations for the ring and inclined plane. [3p]

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