A rigid massess rod has three particles with equal masses attached to it as show

Question

A rigid massess rod has three particles with equal masses attached to it as shown belO·The rod iS free to rotate in a vertical plane about a frictionless a le perpendicular to the rod through the point and is released rro position att0, ASsume that m and d are known. (Use the following as necessary:m, d, and g.) est in the horizontal (a) Find the moment of inertia of the system of three particles about the pivat. (b) Find the torque acting on the system at t 0. () Find the angular acceleration of the system att-o. magnitude direction (d) Find the inear acceleration of the partide labeled 3 att- o. magnitude direction (e) Find the maximum kinetic energy of the system f) Find the maximum angular speed attained by the rod g) Find the maximum angular mcmentum of the system (h) Find the maximum speed attained by the partidle labeled 2

Explanation / Answer

(a) IP = mr² = m*(d/3)² + m*(2d/3)² + m*(4d/3)² = 21md² / 9 = 7md² / 3

(b) = (m*4d/3 + m*d/3 - m*2d/3)*g = mdg

(c) = / IP = mdg / (7md²/3) = 3g / 7d

(d) a = r = (3g/7d)*2d/3 = 2g / 7

(e) Relative to a vertical position, the system has
PE = m*(4d/3 + d/3 - 2d/3)g = mdg
so that's the maximum KE (achieved when the rod is vertical).

(f) KE = ½*IP*²
mdg = ½ * (7md² / 3) * ²
g = (7d / 6) * ²
6g / 7d = ²
max = (6g / 7d)

(g) max L = IP * = (7md² / 3) * (6g / 7d) = md² * (14g / 3d)

(h) max v2 = *d/3 = (d/3) * (6g / 7d) = (2gd / 21)

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