Time Speed And Distance

Let's analyze each statement regarding a system of N interacting particles (like a protein molecule):

1. Statement (a): The motion of the system can be split into translational, rotational, and vibrational motions.
   • This is a standard approach in physics and chemistry for analyzing the dynamics of multi-particle systems. The overall motion can be described by the translation of the center of mass, the rotation of the system around the center of mass, and the internal vibrations of the particles relative to each other.
   • This statement is generally TRUE.
2. Statement (d): Number of vibrational degrees of freedom are 3.
   • The translational motion of the entire system is described by the movement
   • These 3N degrees of freedom are distributed among translation, rotation, and vibration.
   • Number of Vibrational Degrees of Freedom = Total DOF - Translational DOF - Rotational DOF
   • Number of Vibrational Degrees of Freedom = 3N - 3 - 3 = 3N - 6 (for a non-linear system).
   • The statement claims the number of vibrational degrees of freedom is 3. This is only true if 3N - 6 = 3, which means 3N = 9, or N = 3. For any system with N ≠ 3 particles, the number of vibrational degrees of freedom is not 3. For a protein molecule, N is large, so 3N-6 is much larger than 3.
   • Statement (d) is FALSE in general.
3. Evaluate Statement (e): Conservation Laws
   • An isolated system is one with no external forces or torques acting on it.
   • Conservation of Energy: In the absence of external work and non-conservative internal forces, the total energy (kinetic + potential) of an isolated system is conserved.
   • Conservation of Angular Momentum: In the absence of external torques, the total angular momentum of an isolated system is conserved.
   • These are fundamental principles of physics.
   • Statement (e) is TRUE.
4. Conclusion: Statement (c) makes a claim about the number of vibrational degrees of freedom that is generally incorrect for a system of N particles.

The FALSE statement is (c).

1. Define Rates:
   • Let , , and be the rates at which pipes A, B, and C fill the tank, respectively (measured in fraction of tank filled per hour).
   • The time taken for a pipe alone is the reciprocal of its rate (e.g., time for C alone is ).
2. Information from the First Statement: "Three pipes A, B, and C can fill a tank in 6 hrs."
   • This means their combined rate fills 1 tank in 6 hours.
   • Combined rate = .
   • Equation: (where 1 represents the whole tank).
   • Therefore, (Equation 1).
3. Information from the Second Statement: "After working at it together for 2 hrs, C is closed and A and B can fill the remaining part in 7 hrs."
   • Work done in the first 2 hours: Pipes A, B, and C work together.

     Using Equation 1, Work = of the tank.

   • Remaining work:

   • Work done by A and B: Pipes A and B fill the remaining of the tank in 7 hours.

     Combined rate of A and B = .

   • Therefore, (Equation 2).
4. Find the Rate of C ( ):
   • We have Equation 1:
   • We have Equation 2:
   • Substitute the value of from Equation 2 into Equation 1:

   • Solve for :

   • Find a common denominator (42):

     So, pipe C fills of the tank per hour.

5. Calculate Time for C Alone:
   • The time taken by C alone is the reciprocal of its rate:

6. Compare with Options: The calculated time is 14 hours.

The correct option is (a) 14.