Question

Two wheels of moment of inertia 4 kgm² rotate side by side at the rate of 120 rev / min and 240 rev / min respectively in the opposite directions. If now both the wheels arc coupled by means of weigntless shaft so that both the wheels now rotate with a common angular speed, find the new speed of rotation. (Ans: 60 r.p.m.)

Answer 1

external torque acts on system of wheels is zero. So, angular momentum is conserved.
e.g., initial angular momentum = final angular momentum
$I_1\omega_1+I_2\omega_2=(I_1+I_2)\omega$

given, $I_1=I_2=4kgm^2$
$\omega_1=120 rpm$
$\omega_2=-240rpm$ [opposite direction of first wheel]

now,
4 × 120 - 4 × 240 = (4 + 4)$\omega$
$\omega=-60rpm$
Here negative sign show that angular velocity is in direction of 2nd wheel.
so, angular speed = 60rpm

Answer 2

Answer:

The momentum of Inertia of wheels =4 kg m

2

Angular speed before coupling :- w

1

=240 rev min

−1

w

2

=120 rev min

−1

Let w be the angular speed of wheels after coupling

As no torque is applied, angular momentum is conserved

Angular Momentum Before Coupling = Angular Momentum After Coupling

Iw

1

−Iw

2

=Iw+Iw

opp direction

Δ(240)−Δ(120)=Δ(2w)

8w=Δ80

w=60 rev/min