A round object of radius R and mass M and center of mass moment of inertia I, with its center of mass at its geometric center is placed on an inclined plane as shown and released from rest. It rolls without slipping down a length l along the ramp, during which time the height of the center of mass decreases by h, as shown.
Rigid Body
A rigid body is an object which does not change its shape or undergo any significant deformation due to an external force or movement. Mathematically speaking, the distance between any two points inside the body doesn't change in any situation.
Rigid Body Dynamics
Rigid bodies are defined as inelastic shapes with negligible deformation, giving them an unchanging center of mass. It is also generally assumed that the mass of a rigid body is uniformly distributed. This property of rigid bodies comes in handy when we deal with concepts like momentum, angular momentum, force and torque. The study of these properties – viz., force, torque, momentum, and angular momentum – of a rigid body, is collectively known as rigid body dynamics (RBD).
Write down the initial energy Ei of the system, using the horizontal
dashed line as the zero height for the gravitational potential energy.
Write down the final energy Ef of the system in terms of the translational velocity v of the center of mass, the
Assume rolling without slipping! How are w, v, and R related? Use
that relation to rewrite Ef in terms of v, M, I, and R.
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