A mass m, lying on a horizontal frictionless surface is connected to mass M as shown in Fig. 4.24. The system is now released. The velocity of mass m when mass M as descended a
The Correct Answer and Explanation is:
To solve the problem, we need to determine the velocity of mass mm when mass MM has descended a height hh, assuming a frictionless and ideal pulley system.
System Setup:
- Block mm lies on a horizontal frictionless surface.
- Block MM hangs vertically and is connected to block mm via a light, inextensible string passing over a frictionless pulley.
- When the system is released, MM descends and mm moves horizontally.
- Let vv be the velocity of both blocks after MM has descended a height hh (since they’re connected, both have the same magnitude of velocity).
Using Energy Conservation:
- Initial total mechanical energy:
- Kinetic energy = 0 (since both are at rest)
- Potential energy = MghMgh (since only mass MM changes height)
- Final total mechanical energy:
- Kinetic energy of mass mm: 12mv2\frac{1}{2}mv^2
- Kinetic energy of mass MM: 12Mv2\frac{1}{2}Mv^2
- Potential energy of MM: 0 (taken as reference level)
According to conservation of mechanical energy: Initial Total Energy=Final Total Energy\text{Initial Total Energy} = \text{Final Total Energy} Mgh=12mv2+12Mv2Mgh = \frac{1}{2}mv^2 + \frac{1}{2}Mv^2 Mgh=12(m+M)v2Mgh = \frac{1}{2}(m + M)v^2
Solving for vv: v2=2Mghm+Mv^2 = \frac{2Mgh}{m + M} v=2Mghm+Mv = \sqrt{\frac{2Mgh}{m + M}}
Correct Answer:
Option (c): \boxed{v = \sqrt{\frac{2Mgh}{m + M}}
Explanation:
This result arises from applying the principle of conservation of energy. Since the system is ideal (no friction or energy loss), the potential energy lost by the falling mass MM is completely converted into the kinetic energy of both masses. The key step is to equate the loss in potential energy to the total kinetic energy gained by both masses, because both are in motion due to the inextensible string.
This approach avoids using Newton’s laws or acceleration, making it an efficient method for such problems.
