Three tiny point masses are placed on an x-y axis. Mass 1 is unknown and it is p

Question

Three tiny point masses are placed on an x-y axis. Mass 1 is unknown and it is placed at x = 0 m, y = +2.58 meters. Mass 2 is 4 kg and is placed at x = -3.59 , y = 0 m.   Mass 3 is 3 kg and is placed at x = 0 , y = -0.79 m. When the masses are spun at 2 rad/s around the origin (the z axis), their rotational kinetic energy about the origin is 570 Joules. What is the rotational kinetic energy of the masses (they do not change location) if they are spun at 2 rad/s about the x axis? Answer in Joules.

Explanation / Answer

x1 =0 , y1 =2.58 m

x2 =-3.59 m , y2 =0

x3 = 0 ,y3 =-0.79 m

m2 =4 kg, m3 = 3 kg

w= 2 rad/s, K =570 J

Moment of inertia about an z axis is I = m1r1^2 +m2r2^2 +m3r3^2

Here r is perpendicular distance from Z axis

Iz = (m1*(2.58)2)+(4*(-3.59)2+(3*(-0.79)2)

Iz = 6.6564m1 + 51.6 +1.87

Iz = 6.6564m1 +53.47

K = (1/2)Iw2

570 = (1/2)[ 6.6564m1 +53.47] (2*2)

231.53 =6.656 m1

m1 = 34.783 kg

Moment of inertia about x- axis is

Moment of inertia about an z axis is Ix = m1r1^2 +m2r2^2 +m3r3^2

Here r is perpendicular distance from x axis

Ix = (34.783*(2.58)2)+(4*(0)2+(3*(-0.79)2)

Ix = 231.53 +0+1.8723 =233.4 kg.m/s

w =2 rad/s

K =(1/2)Iw^2

K = (1/2) (233.4)(2*2)

K =466.8 J

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