To solve the problem of the sand sack and the mass M connected by a string over a pulley, we need to analyze the forces acting on the system and apply the principles of conservation of energy and kinematics. Let's break it down step by step.
Understanding the System
Initially, we have a balanced system with a mass M on one side and a sack containing sand on the other. The sack has a mass of sand that decreases over time as it leaks out. The mass of the empty sack is given as M/5. When one-fourth of the sand is left, we need to determine the velocity of the sack.
Initial Setup
- Let the initial mass of the sand be denoted as m.
- The total initial mass of the sack with sand is m + M/5.
- As sand leaks out at a constant rate B, the mass of the sand decreases over time.
Mass Calculation
When one-fourth of the sand is left, the mass of the sand can be expressed as:
m_remaining = (3/4)m
The total mass of the sack at this point becomes:
m_remaining + M/5 = (3/4)m + M/5
Applying Newton's Second Law
As the sand leaks out, the forces acting on the system change. We can apply Newton's second law to analyze the motion of the sack. The net force acting on the sack will be the difference between the gravitational force acting on the mass M and the gravitational force acting on the sack with the remaining sand.
Force Analysis
The gravitational force acting on mass M is:
F_M = M * g
The gravitational force acting on the sack with the remaining sand is:
F_sack = (3/4)m + M/5) * g
Net Force and Acceleration
The net force F_net acting on the system can be expressed as:
F_net = F_M - F_sack
Substituting the forces, we have:
F_net = M * g - ((3/4)m + M/5) * g
Factoring out g, we get:
F_net = g * (M - (3/4)m - M/5)
Using Conservation of Energy
As the sack descends, it converts potential energy into kinetic energy. The change in potential energy as the sack descends a height h can be equated to the kinetic energy gained by the sack:
mgh = (1/2)mv^2
Where v is the velocity of the sack. Rearranging gives:
v = sqrt(2gh)
Finding the Height
The height h can be related to the mass of sand lost. If the sand leaks out at a constant rate B, we can express the time t it takes for one-fourth of the sand to leak out as:
t = (1/4)m / B
During this time, the sack will have descended a height h which can be calculated based on the acceleration derived from the net force.
Final Calculation
By substituting the values of mass and height into the equations derived, we can find the velocity of the sack when one-fourth of the sand is left. The final expression will depend on the specific values of m, M, and B.
In summary, the velocity of the sack when one-fourth of the sand is left can be determined by analyzing the forces acting on the system, applying Newton's laws, and using the principles of energy conservation. This approach allows us to derive a clear relationship between the mass of the sand, the gravitational forces, and the resulting motion of the sack.
