Question

Rigid rods of negligible mass lying along the y axis connect three particles. The system rotates about the x axis with an angular speed of 2.40 rad/s. An x y plane is shown. Three particles are all connected to each other by two rigid rods that are along the y-axis. The 4.00 kg particle is located at y = 3.00 m. The 2.00 kg particle is located at y = −2.00 m. The 3.00 kg mass is located at y = −4.00 m. The axis of rotation is the x-axis and it rotates in a counterclockwise direction (from the perspective of the side, along the +x-axis).

(a) Find the moment of inertia about the x axis. kg · m2

(b) Find the total rotational kinetic energy evaluated from 1 2 Iω2. J

(c) Find the tangential speed of each particle. 4.00 kg particle m/s 2.00 kg particle m/s 3.00 kg particle m/s

(d) Find the total kinetic energy evaluated from 1 2 mivi2 . J

(e) Compare the answers for kinetic energy in parts (b) and (d).

Answer 1

let m1 = 4 kg, y1 = 3.00 m
m2 = 2 kg, y2 = -2 m
m3 = 3 kg, y3 = -4 m

a) I = m1*y1^2 + m2*y2^2 + m3*y3^2

= 4*3^2 + 2*2^2 + 3*4^2

= 92.0 kg.m^2

b) rotational kinetic energy, KE = (1/2)*I*w^2

= (1/2)*92*2.4^2

= 265 J

c) tangential speed of 4 kg, v1 = 3*2.4 = 7.20 m/s

tangential speed of 2 kg, v2 = 2*2.4 = 4.80 m/s

tangential speed of 3 kg, v3 = 4*2.4 = 9.60 m/s

d) KE = (1/2)*m1*v1^2 + (1/2)*m2*v2^2 + (1/2)*m3*v3^2

= (1/2)*4*7.2^2 + (1/2)*2*4.8^2 + (1/2)*3*9.6^2

= 265 J

e) both answers are same