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Ch 10: Interactions and Potential Energy
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 10: Interactions and Potential EnergyProblema 49a
Capitolo 10, Problema 49a

Two blocks with masses mA and mB are connected by a massless string over a massless, frictionless pulley. Block B, which is more massive than block A, is released from height h and falls. Write an expression for the speed of the blocks just as block B reaches the ground.

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1
Identify the forces acting on the system: Block A experiences tension (T) upward and its weight (mA * g) downward. Block B experiences tension (T) upward and its weight (mB * g) downward. The system accelerates due to the unbalanced force caused by the difference in weights.
Apply Newton's second law to each block. For block A: T - mA * g = mA * a. For block B: mB * g - T = mB * a. Here, 'a' is the acceleration of the system, and 'T' is the tension in the string.
Combine the two equations to eliminate T and solve for the acceleration 'a'. Adding the equations gives: mB * g - mA * g = (mA + mB) * a. Simplify to find: a = (mB - mA) * g / (mA + mB).
Use the kinematic equation to find the final speed of the blocks. Since block B falls a distance h starting from rest, the equation v² = u² + 2 * a * h applies, where u = 0 (initial velocity), a is the acceleration derived earlier, and h is the height. Substitute a = (mB - mA) * g / (mA + mB) into the equation to get: v² = 2 * ((mB - mA) * g / (mA + mB)) * h.
Take the square root of both sides to find the final expression for the speed: v = sqrt(2 * ((mB - mA) * g * h) / (mA + mB)). This is the speed of both blocks just as block B reaches the ground.

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Concetti chiave

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Conservation of Energy

The principle of conservation of energy states that in a closed system, the total energy remains constant. In this scenario, the potential energy of block B at height h is converted into kinetic energy as it falls. This relationship allows us to equate the initial potential energy to the final kinetic energy to find the speed of the blocks.
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Conservation Of Mechanical Energy

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, defined mathematically as KE = 1/2 mv², where m is the mass and v is the velocity. In this problem, as block B falls, its kinetic energy increases while block A also moves, and understanding this relationship is crucial for deriving the speed of the blocks when block B reaches the ground.
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Intro to Rotational Kinetic Energy

Acceleration due to Gravity

Acceleration due to gravity (g) is the acceleration experienced by an object in free fall near the Earth's surface, approximately 9.81 m/s². This constant influences the motion of both blocks in the system, affecting how quickly block B falls and, consequently, the speed of both blocks when block B reaches the ground.
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Acceleration Due to Gravity
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