Question

Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc $D_1 $ has 2 kg mass and 0.2 m radius and initial angular velocity of $ 50\, rad \,s^{-1}.$ Disc $ D_2 $ has 4 kg mass, 0.1 m radius and initial angular velocity of $ 200\, rad\, s^{-1} $. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad $ s^{-1}$ ) of the system is

• A. 60
• B. 100
• C. 120
• D. 40

Answer 1

The Correct Option is B

This problem involves using the principle of conservation of angular momentum. When two rotating discs are brought into contact and allowed to rotate together, their individual angular momenta before contact remain conserved and they rotate with a common angular velocity after contact. We will solve this step-by-step.

1. The moment of inertia I for a disc rotating about its axis is given by I = \frac{1}{2} m r^2, where m is the mass and r is the radius.
2. Calculate the moment of inertia for each disc:
   • For Disc D_1: I_1 = \frac{1}{2} \times 2\, \text{kg} \times (0.2\, \text{m})^2 = 0.04\, \text{kg} \cdot \text{m}^2
   • For Disc D_2: I_2 = \frac{1}{2} \times 4\, \text{kg} \times (0.1\, \text{m})^2 = 0.02\, \text{kg} \cdot \text{m}^2
3. Using the conservation of angular momentum, the initial angular momentum of the system is equal to the final angular momentum, i.e., I_1 \omega_1 + I_2 \omega_2 = (I_1 + I_2) \omega_f, where \omega_1 and \omega_2 are the initial angular velocities, and \omega_f is the final angular velocity.
4. Substitute the given values:
   • I_1 \omega_1 = 0.04\, \text{kg} \cdot \text{m}^2 \times 50\, \text{rad/s} = 2\, \text{kg} \cdot \text{m}^2/\text{s}
   • I_2 \omega_2 = 0.02\, \text{kg} \cdot \text{m}^2 \times 200\, \text{rad/s} = 4\, \text{kg} \cdot \text{m}^2/\text{s}
   • Therefore, 2 + 4 = (0.04 + 0.02) \omega_f
5. Solve for \omega_f: 6 = 0.06 \omega_f, hence \omega_f = \frac{6}{0.06} = 100\, \text{rad/s}.

Therefore, the final angular velocity of the system is 100 rad/s. Hence, the correct answer is 100.