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Ch. 08 - Conservation of Energy
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 08 - Conservation of EnergyProblem 22c
Chapter 8, Problem 22c

Two masses are connected by a string as shown in Fig. 8–35. Mass mA = 3.5 kg rests on a frictionless inclined plane, while mB = 5.0 kg is initially held at a height of h = 0.75 m above the floor. Use conservation of energy to find the velocity of the masses just before mB hits the floor. You should get the same answer as in part (b).

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Identify the system: The two masses (m_A and m_B) are connected by a string, and the system is influenced by gravity and the inclined plane. Since the plane is frictionless, no energy is lost to friction. The problem involves using the principle of conservation of energy.
Write the total mechanical energy of the system at the initial state: The initial energy consists of the gravitational potential energy of m_B (m_B * g * h) and the gravitational potential energy of m_A due to its position on the inclined plane. Since m_A is not moving initially, its kinetic energy is zero.
Write the total mechanical energy of the system at the final state: Just before m_B hits the floor, m_B has lost all its gravitational potential energy, and both masses have gained kinetic energy. The kinetic energy of each mass is (1/2) * m * v^2, where v is the velocity of the masses (they share the same velocity due to the string). Additionally, m_A has moved up the incline, gaining gravitational potential energy (m_A * g * h_A, where h_A is the vertical height it has risen).
Apply the conservation of energy principle: Set the total initial energy equal to the total final energy. This gives the equation: m_B * g * h + m_A * g * h_A_initial = (1/2) * m_A * v^2 + (1/2) * m_B * v^2 + m_A * g * h_A_final. Simplify the equation by noting that h_A_final = d * sin(θ), where d is the distance m_A has moved along the incline and θ is the angle of the incline.
Solve for the velocity v: Rearrange the equation to isolate v. Combine terms involving v^2, and substitute known values for m_A, m_B, g, h, and θ. This will yield an expression for v, the velocity of the masses just before m_B hits the floor.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant over time. In this scenario, the potential energy of mass m_B at height h is converted into kinetic energy as it falls. This concept is crucial for analyzing the motion of the masses and determining their velocities just before impact.
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Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is the mass and v is the velocity. In this problem, as mass m_B descends, its potential energy decreases while its kinetic energy increases, allowing us to find the velocity of both masses just before m_B hits the floor.
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Potential Energy

Potential energy is the stored energy of an object due to its position or configuration, commonly expressed as PE = mgh, where m is mass, g is the acceleration due to gravity, and h is height. In this case, the potential energy of mass m_B at height h is converted into kinetic energy as it falls, which is essential for applying the conservation of energy principle.
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