To solve this problem, we need to analyze the dynamics of the system as the sand leaks out of the sack. We have a massless, inextensible string, a pulley with mass, and a sack of sand that loses mass over time. Let's break down the components and apply the principles of physics to find the velocity of the sack when one-fourth of the sand remains.
Understanding the System
Initially, the system is balanced, meaning the weight of the sand-filled sack equals the weight of the mass m on the other side of the pulley. As the sand leaks out, the mass of the sack decreases, which will affect the forces acting on the system.
Defining the Variables
- Mass of the empty sack: m/5
- Initial mass of the sand: Let’s denote the initial mass of the sand as M.
- Mass of the sack when one-fourth of the sand is left: (3/4)M + m/5
- Rate of sand leaking: X (constant rate)
- Time taken to leak one-fourth of the sand: T = (1/4)M / X = M/(4X)
Calculating the Masses
When one-fourth of the sand is left, the mass of the sand is (3/4)M. Therefore, the total mass of the sack at this point is:
Mass of the sack = (3/4)M + (m/5)
Applying Newton's Second Law
To find the velocity of the sack, we can apply Newton's second law. The forces acting on the sack and the mass m will determine the acceleration of the system.
Forces Acting on the Sack
The downward force on the sack is due to its weight:
Weight of the sack = (3/4)M + (m/5) * g
The upward force is the tension in the string, which we will denote as T.
Forces Acting on the Mass m
The downward force on mass m is:
Weight of mass m = m * g
Since the system is accelerating, we can set up the equations:
For the Sack:
(3/4)M + (m/5) * g - T = (3/4)M + (m/5) * a
For the Mass m:
m * g - T = m * a
Solving the Equations
From the second equation, we can express T:
T = m * g - m * a
Substituting T into the first equation gives us:
(3/4)M + (m/5) * g - (m * g - m * a) = (3/4)M + (m/5) * a
Rearranging this equation will allow us to isolate the acceleration a. After some algebra, we can find the relationship between the acceleration of the sack and the mass of the sand.
Finding the Velocity
Once we have the acceleration, we can find the velocity of the sack after the time T:
v = a * T
Substituting T = M/(4X) into the equation will yield the final velocity of the sack when one-fourth of the sand is left.
Conclusion
By following these steps and performing the calculations, you will arrive at the velocity of the sack at the specified moment. This problem illustrates the principles of dynamics and the effect of changing mass on a system in motion. If you need further clarification on any of the steps or concepts, feel free to ask!
