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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: 9780137488179당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Giancoli Douglas 5th editionCh. 08 - Conservation of Energy문제 23a
8장, 문제 23a

The 9.0-kg mass in Fig. 8–36 is held just barely in contact with a spring for which k = 450 N/m . When that mass is released, it falls, compressing the spring and pulling the 3.0-kg mass up. How far does the 9.0-kg mass fall before momentarily coming to rest? Ignore friction in the pulley.
Diagram showing a pulley system with a 9.0 kg mass compressing a spring and a 3.0 kg mass suspended below.

검증된 단계별 안내
1
Identify the key concepts involved: This problem involves energy conservation (gravitational potential energy, elastic potential energy) and the relationship between the two masses connected by the pulley system. The spring constant (k) and the masses are given, and we are tasked with finding the distance the 9.0-kg mass falls before coming to rest.
Set up the energy conservation equation: The total mechanical energy of the system is conserved. Initially, the 9.0-kg mass has gravitational potential energy, and the spring has no elastic potential energy. At the point where the 9.0-kg mass momentarily comes to rest, all the gravitational potential energy lost by the 9.0-kg mass is converted into elastic potential energy of the spring. The equation is: m1gh = 12kx2
Relate the displacement of the two masses: Since the two masses are connected by a pulley, the distance the 9.0-kg mass falls (h) is equal to the distance the spring is compressed (x). Thus, h = x. Substitute this into the energy conservation equation.
Solve for the displacement (x): Substitute the known values into the equation. The mass of the 9.0-kg object is m₁ = 9.0 kg, the spring constant is k = 450 N/m, and g = 9.8 m/s². The equation becomes: 9.0 × 9.8 × x = 12 × 450 × x2
Simplify and solve the quadratic equation: Rearrange the equation into standard quadratic form: 225x2 - 88.2x = 0 . Factorize or use the quadratic formula to solve for x. Discard any negative solution, as distance cannot be negative.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
6m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. In this scenario, the spring constant (k = 450 N/m) will determine how much the spring compresses when the 9.0-kg mass falls.
추천 영상:
가이드 코스
05:27
Spring Force (Hooke's Law)

Gravitational Potential Energy

Gravitational potential energy (U) is the energy an object possesses due to its position in a gravitational field, calculated as U = mgh, where m is mass, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the height. As the 9.0-kg mass falls, its potential energy is converted into kinetic energy and then into spring potential energy when the spring is compressed.
추천 영상:
가이드 코스
06:35
Gravitational Potential Energy

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this problem, the gravitational potential energy of the falling mass is converted into the elastic potential energy of the compressed spring, allowing us to set up an equation to find the distance the mass falls before momentarily coming to rest.
추천 영상:
가이드 코스
06:24
Conservation Of Mechanical Energy
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