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Step-by-Step Detailed Solution
Step 1: Understand the Problem
A uniform chain of length 3 m and total mass 3 kg is placed on a smooth (frictionless) table such that 2 m of the chain is on the table, and 1 m of the chain is hanging off the edge. We need to find the kinetic energy (K) of the chain at the instant it completely slips off the table. We assume the acceleration due to gravity is 10\, \text{m/s}^2 .
Step 2: Identify Initial and Final States
Initial State: 2 m of the chain on the table (at the reference height), 1 m of the chain hanging below the table.
Final State: The entire 3 m of the chain has just slipped off the table and is in motion, all below the table's top level.
Step 3: Assign a Reference for Potential Energy
Let the top of the table be our zero potential energy reference ( U = 0 at the level of the table). Any segment of the chain below this reference will have negative potential energy.
Step 4: Calculate the Initial Potential Energy ( U_i )
• Mass per meter of the chain = \dfrac{3\,\text{kg}}{3\,\text{m}} = 1\,\text{kg/m}.
• The portion off the table is 1 m long, so its mass is 1\,\text{kg}.
• For a uniform segment of length 1 m, its center of mass is at the midpoint, i.e., 0.5\,\text{m} below the table.
Therefore, initial potential energy of the hanging part:
U_i = - \bigl(\text{mass of hanging part}\bigr)\,g\bigl(\text{distance of COM below table}\bigr).
Plugging in the values:
U_i = - (1\,\text{kg})(10\,\text{m/s}^2)(0.5\,\text{m}) = -5\,\text{J}.
The portion on the table is at the reference height, so its potential energy contribution is zero. Thus,
U_i = -5\,\text{J}.
Step 5: Calculate the Final Potential Energy ( U_f )
Once the chain completely slips off the table, the entire chain of length 3 m is below the table-top level. The center of mass of a uniform chain of length 3 m lies at its midpoint (1.5 m from either end), so it is 1.5\,\text{m} below the table.
Thus, the entire mass is 3\,\text{kg} , and the potential energy now is:
U_f = - \bigl(3\,\text{kg}\bigr) \bigl(10\,\text{m/s}^2\bigr) \bigl(1.5\,\text{m}\bigr)
= -45\,\text{J}.
Step 6: Use Conservation of Energy to Find Kinetic Energy
Because the table is smooth (no friction), mechanical energy is conserved. The loss in potential energy of the chain is gained as its kinetic energy ( K ). Hence,
\Delta U = U_f - U_i = K.
Substitute the values:
\Delta U = (-45)\,\text{J} - (-5)\,\text{J} = -40\,\text{J}.
The negative sign indicates a decrease in potential energy, which appears as a positive gain in kinetic energy. Therefore,
K = 40\,\text{J}.
Step 7: Final Answer
The kinetic energy of the chain as it completely slips off the table is 40\,\text{J} .
