Chapter 7: Problem 38
Two blocks with different masses are attached to either end of a light rope that passes over a light, frictionless pulley suspended from the ceiling. The masses are released from rest, and the more massive one starts to descend. After this block has descended 1.20 m, its speed is 3.00 m/s. If the total mass of the two blocks is 22.0 kg, what is the mass of each block?
Short Answer
Step by step solution
Understand the Problem
Apply the Law of Conservation of Mechanical Energy
Express Masses Using Total Mass Constraint
Substitute Masses into the Energy Equation
Solve for the Mass of \(m_2\)
Calculate Numerical Solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mechanical Energy Conservation
When the blocks are released from rest, the potential energy due to the height of a block is converted into kinetic energy as it descends. This principle is mathematically expressed as:
- Potential energy lost = Kinetic energy gained.
Frictionless Pulley System
This simplification leads to reliable energy transitions between the blocks in the system. The pulley merely changes the direction of the force applied by gravity, causing the heavier block to descend while lifting the lighter one. The lack of friction ensures that no mechanical energy is lost to heat or sound. This setup allows us to focus solely on the relationship between masses, speed, and acceleration, making it a perfect real-world application of theoretical physics principles.
Mass Calculation
- Let the heavier block be represented as \(m_1\) and the lighter as \(m_2\):
\(m_1 + m_2 = 22\, \text{kg}\).
- A crucial step is calculating the balance between the given total mass and the speed reached after a given descent.
Kinematics Equations
- Initial speed \(u\): 0 \(\text{m/s}\), as blocks start from rest.
- Final speed \(v\): 3 \(\text{m/s}\), given.
- Distance \(s\): 1.20 \(\text{m}\), over which the mass descends.