Consider a "round" rigid body with moment of inertia I = BMR2, where M is the body's mass, R is the body's radius, and ß is a constant depending on the type of the body. Rigid Body Uniform hoop Uniform solid cylinder Uniform spherical shell Uniform solid sphere 1 1/2 R- 2/3 2/5 k The center of the "round" rigid body is attached to a spring of force constant k, and then the body is made to roll without slipping on a rough horizontal surface. Due to the spring, it is expected the body will oscillate by rolling back and forth from its resting position.
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).
Consider a “round” rigid body with moment of inertia I = BMR2, where M is the body’s mass, R is the body’s radius, and B (beta) is a constant depending on the type of the body.
The center of the “round” rigid body is attached to a spring of force constant k, and then the body is made to roll without slipping on a rough horizontal surface. Due to the spring, it is expected the body will oscillate by rolling back and forth from its resting position.
A. Determine the angular frequency and the period for small oscillations of the round rigid body. Express your answers in terms of B (beta).
B. Among the four “round” rigid bodies shown at the table, for the same masses and radii, which among them will have the most number of cycles per second.
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