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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: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 10: Interactions and Potential EnergyProblem 49a
Chapter 10, Problem 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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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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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 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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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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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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Related Practice
Textbook Question

You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 10-m-high hill, then descends 15 m to the track's lowest point. You've determined that the spring can be compressed a maximum of 2.0 m and that a loaded car will have a maximum mass of 400 kg. For safety reasons, the spring constant should be 10% larger than the minimum needed for the car to just make it over the top. What is the maximum speed of a 350 kg car if the spring is compressed the full amount?

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Textbook Question

A horizontal spring with spring constant 100 N/m is compressed 20 cm and used to launch a 2.5 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Use work and energy to find how far the box slides across the rough surface before stopping.

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Textbook Question

A freight company uses a compressed spring to shoot 2.0 kg packages up a 1.0-m-high frictionless ramp into a truck, as FIGURE P10.52 shows. The spring constant is 500 N/m and the spring is compressed 30 cm. What is the speed of the package when it reaches the truck?

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Textbook Question

A block of mass m slides down a frictionless track, then around the inside of a circular loop-the-loop of radius R . From what minimum height h must the block start to make it around without falling off? Give your answer as a multiple of R.

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Textbook Question

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. A 1.0 kg block and a 2.0 kg block are connected by a massless string over a massless, frictionless pulley. The impact speed of the heavier block, after falling, is 1.8 m/s. From how high did it fall?

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The ice cube is replaced by a 50 g plastic cube whose coefficient of kinetic friction is 0.20. How far will the plastic cube travel up the slope? Use work and energy.

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