Consider the simple pendulum: a ball hanging at the end of a string. Derive the expression for the period of this physical pendulum, taking into account the finite sizeball (i.e. the ball is not a point mass). Assume that the string is massless. Start with the expression for the period T'of a physical pendulum with small amplitude oscillatiT= 2πThe moment of inertia of the ball about an axis through the center of the ball isHere, I, is the moment of inertia about an axis through the pivot (fixed point at the top of the string, m is the mass of the ball, g is the Earth's gravitational constant ofacceleration, and h is the distance from the pivot at the top of the string to the center of mass of the ball.Note, this pre-lab asks you to do some algebra, and may be a bit tricky.ImghIball = / mr²T If we take the limit for the pendulum where L » r, what will the period be?Note: the symbol >> means "much greater than." This is a common notation used in physics, which means that anywhere we see the sum of the two variables in theexpression (L and r) we can ignore the smaller variable. For example, in the limit where x ≫y, the expression 2x + 3y becomes simply 3y.T= 2T.OOT = 2x√√2T2L5gI = 2√²/²7T5gLT = 27.5gL√T = 27.9○ T = 2x√ √2+²5L5g2r5g

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If we take the limit for the pendulum where L » r, what will the period be? Note: the symbol >> means "much greater than." This is a common notation used in physics, which means that anywhere we see the sum of the two variables in the expression (L and r) we can ignore the smaller variable. For example, in the limit where x >y, the expression 2x + 3y becomes simply 3y. 2L : 2 V 59 T = 77² T = 2√ 5gL T = 2π T= 2T₁ 272 5gL L 9 2r²L T= 2√ 59 2r T = 2π√√ 5g

If we take the limit for the pendulum where L » r, what will the period be? Note: the symbol >> means "much greater than." This is a common notation used in physics, which means that anywhere we see the sum of the two variables in the expression (L and r) we can ignore the smaller variable. For example, in the limit where x >y, the expression 2x + 3y becomes simply 3y. 2L : 2 V 59 T = 77² T = 2√ 5gL T = 2π T= 2T₁ 272 5gL L 9 2r²L T= 2√ 59 2r T = 2π√√ 5g

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter11: Dynamics Of Rigid Bodies

Section: Chapter Questions

Problem 11.15P

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