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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 38b
8장, 문제 38b

A spring ( k = 75 N/m) has an equilibrium length of 1.00 m. The spring is compressed to a length of 0.50 m and a mass of 2.0 kg is placed at its free end on a frictionless slope which makes an angle of 41° with respect to the horizontal (Fig. 8–41). The spring is then released. If the mass is attached to the spring, how far up the slope will the mass move before coming to rest?
A compressed spring on a frictionless slope, with a mass at its end, angled at 41° and compressed to 0.50 m.

검증된 단계별 안내
1
Determine the potential energy stored in the compressed spring using the formula for elastic potential energy: \( U_s = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the compression distance. Here, \( x = 1.00 \ \text{m} - 0.50 \ \text{m} = 0.50 \ \text{m} \).
Recognize that the spring's potential energy will be converted into gravitational potential energy as the mass moves up the slope. The gravitational potential energy is given by \( U_g = m g h \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the vertical height gained by the mass.
Relate the vertical height \( h \) to the distance \( d \) traveled along the slope using the angle \( \theta \): \( h = d \sin \theta \). Substitute this into the gravitational potential energy formula to express \( U_g \) in terms of \( d \): \( U_g = m g d \sin \theta \).
Set the elastic potential energy equal to the gravitational potential energy to find the distance \( d \): \( \frac{1}{2} k x^2 = m g d \sin \theta \). Rearrange this equation to solve for \( d \): \( d = \frac{\frac{1}{2} k x^2}{m g \sin \theta} \).
Substitute the known values into the equation: \( k = 75 \ \text{N/m} \), \( x = 0.50 \ \text{m} \), \( m = 2.0 \ \text{kg} \), \( g = 9.8 \ \text{m/s}^2 \), and \( \sin \theta = \sin 41^\circ \). This will give the distance \( d \) the mass moves up the slope before coming to rest.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
8m

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's compression will generate a force that acts on the mass when released, influencing its motion up the slope.
추천 영상:
가이드 코스
05:27
Spring Force (Hooke's Law)

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 potential energy stored in the compressed spring will convert into kinetic energy as the mass moves up the slope, and eventually into gravitational potential energy at the highest point of its ascent.
추천 영상:
가이드 코스
06:24
Conservation Of Mechanical Energy

Inclined Plane Dynamics

When analyzing motion on an inclined plane, the forces acting on the mass include gravitational force, normal force, and the spring force. The component of gravitational force acting parallel to the slope affects how far the mass will move up the incline before coming to rest, requiring the use of trigonometric functions to resolve forces based on the angle of the slope.
추천 영상:
가이드 코스
06:59
Intro to Inclined Planes
관련 실천
교과서 질문

A spring ( k = 75 N/m) has an equilibrium length of 1.00 m. The spring is compressed to a length of 0.50 m and a mass of 2.0 kg is placed at its free end on a frictionless slope which makes an angle of 41° with respect to the horizontal (Fig. 8–41). The spring is then released. Now the incline has a coefficient of kinetic friction μₖ. If the block, attached to the spring, is observed to stop just as it reaches the spring’s equilibrium position, what is the coefficient of friction μₖ?

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교과서 질문

Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine the velocity of the block at point C.

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교과서 질문

Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine the thermal energy produced as the block slides from B to C.

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교과서 질문

Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine the stiffness constant k for the spring.

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