Find the Angular velocity and period of oscillations of a solid sphere of mass m and radius R about a point on its surface book of Marion,Stephen&Thornton, Jerry (Classical Dynamics of Particles and Systems) by v (units: vibrations per unit time or Hertz, Hz,cumes a will befor brevity, although "angular frequency" is to be understood.13.2 SIMPLE HARMONIC OSCILLATOR10EXAMPLE 3.1Find the angular velocity and period of oscillation of a solid sphere of mass mand radius Rabout a point on its surface. See Figure 3-1.Solution. Let the rotational inertia of the sphere be Iabout the pivot point. In ele-mentary physics we learn that the value of the rotational inertia about an axisthrough the sphere's center is 2/5 mR. If we use the parallel-axis theorem, the rota-tional inertia about the pivot point on the surface is 2/5 mR + mR = 7/5 mR.The equilibrium position of the sphere occurs when the center of mass (center ofsphere) is hanging directly below the pivot point. The gravitational force F= mgpulls the sphere back towards the equilibrium position as the sphere swings backand forth with angle 0. The torque on the sphere is N= la, where a - ö is the an-gular acceleration. The torque is also N = R x F, with N= RFsin 0 = Rmg sin 8.For small oscillations, we have N- Rmg 0. We must have 1ö = -Rmg 0 for theequation of motion in this case, because as 8 increases, ở is negative. We need tosolve the equation of motion for 6.RmgThis equation is similar to Equation 3.5 and has solutions for the angular fre-quency and period from Equations 3.14 and 3.15,RmgRmg5gVTR7and= 27\1= 27 Rmg- 27N 5gRmgPivotRFIGURE 3-1 Example 3.1. The physical pendulum (sphere).1043/ OSCILLATIONSNote that the mass m does not enter. Only the distance R to the center ofmass determines the oscillation frequency.3.3 Harmonic Oscillations in Two DimensionsWe next consider the motion of a particle that is allowed two degrees of freedom.We take the restoring force to be proportional to the distance of the particle froma force center located at the origin and to be directed toward the origin:F = -kr(3.16)which can be resolved in polar coordinates into the componentsF, = - kr cos 8 = - kxF, = - kr sin e = – ky S(3.17)%3DThe equations of motion are

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Find the Angular velocity and period of oscillations of a solid sphere of mass m and radius R about a point on its surface book of Marion,Stephen&Thornton, Jerry (

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter7: Hamilton's Principle-lagrangian And Hamiltonian Dynamics

Section: Chapter Questions

Problem 7.1P

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