Answer:
a) the final angular speed is 0.738 rad/s
b) the change in kinetic energy = 0.3 J
Explanation:
the two 1 kg objects have a total mass of 2 x 1 = 2 kg
radius of rotation of the objects = 0.9 m
moment of inertial of the student and the chair = 6 kg-m^2
initial angular speed of rotation of the sitting student and object system ω1 = 0.61 rad/s
final angular speed of rotation of the sitting student and object system ω2 = ?
moment of inertia of the rotating object is
[tex]I = mr^{2}[/tex] = 2 x [tex]0.9^{2}[/tex] = 1.62 kg-m^2
total moment of inertia of sitting student and object system will be
==> 6 + 1.62 = 7.62 kg-m^2
The initial angular momentum of the sitting student and object system will be calculated from
==> Iω1 = 7.62 x 0.61 = 4.65 kg-rad/s-m^2
if the radius of rotation of the object is reduced to 0.39 m,
new moment of inertia of the rotating object will be
[tex]I = mr^{2}[/tex] = 2 x [tex]0.39^{2}[/tex] = 0.304 kg-m^2
new total moment of inertia of the sitting student and object system will be
==> 6 + 0.304 = 6.304 kg-m^2
The final momentum of the sitting student and object system will be calculated from
==> Iω2 = 6.304 x ω2 = 6.304ω2
According to conservation of angular momentum, initial momentum of the system must be equal to the final momentum of the system. Therefore,
4.65 = 6.304ω2
ω2 = 4.65/6.30 = 0.738 rad/s
b) Rotational kinetic energy of the system = [tex]\frac{1}{2} Iw^{2}[/tex]
for the initial conditions, kinetic energy is
==> [tex]\frac{1}{2} Iw1^{2}[/tex] = [tex]\frac{1}{2}* 7.62*0.61^{2}[/tex] = 1.417 J
for the final conditions, kinetic energy is
==> [tex]\frac{1}{2} Iw1^{2}[/tex] = [tex]\frac{1}{2}*6.304*0.738^{2}[/tex] = 1.717 J
change in kinetic energy = final KE - initial KE
==> 1.717 - 1.417 = 0.3 J
