Ch. 05 - Using Newton's Laws: Friction, Circular Motion, Drag Forces

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 05 - Using Newton's Laws: Friction, Circular Motion, Drag Forces

문제 9

5장, 문제 9

Show that if a skier moves at constant speed straight down a slope of angle θ (Example 5–6), then the coefficient of kinetic friction between skis and snow is μₖ = tanθ. Ignore air resistance.

검증된 단계별 안내

Start by analyzing the forces acting on the skier. The skier is moving at a constant speed, which means the net force along the slope is zero. The forces involved are the gravitational force, the normal force, and the kinetic friction force.

Break the gravitational force into components. The component parallel to the slope is given by

F_{g, \(\text{parallel}\)} = mg \, \(\sin\[\theta\)

, and the component perpendicular to the slope is

F_{g, \(\text{perpendicular}\)} = mg \, \(\cos\]\theta\)

, where

m

is the mass of the skier,

g

is the acceleration due to gravity, and

\(\theta\)

is the angle of the slope.

Write the expression for the kinetic friction force. The kinetic friction force is given by

F_k = \(\mu\)_k F_N

, where

\(\mu\)_k

is the coefficient of kinetic friction and

F_N

is the normal force. Since the normal force balances the perpendicular component of gravity,

F_N = mg \, \(\cos\)\(\theta\)

Set up the condition for constant speed. Since the skier is moving at constant speed, the downhill gravitational force component is exactly balanced by the kinetic friction force. This gives the equation

mg \, \(\sin\[\theta\) = \(\mu\)_k \(\cdot\) mg \, \(\cos\]\theta\)

Simplify the equation to solve for

\(\mu\)_k

. Cancel out

mg

from both sides (since it is nonzero), leaving

\(\sin\[\theta\) = \(\mu\)_k \(\cdot\) \(\cos\]\theta\)

. Divide both sides by

\(\cos\[\theta\)

to get

\(\mu\)_k = \(\tan\]\theta\)

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In the context of the skier, this law helps us understand how the forces acting on the skier (gravity, friction, and normal force) balance out when the skier moves at constant speed, leading to the conclusion about the coefficient of kinetic friction.

Friction and Coefficient of Kinetic Friction

Friction is the force that opposes the relative motion of two surfaces in contact. The coefficient of kinetic friction (μₖ) quantifies this frictional force relative to the normal force. In this scenario, the relationship between the slope angle and the frictional force is crucial, as it allows us to derive that μₖ equals the tangent of the slope angle (tanθ) when the skier moves at constant speed.

Forces on an Inclined Plane

When an object is on an inclined plane, the gravitational force can be resolved into two components: one parallel to the slope, which causes the object to accelerate down the slope, and one perpendicular to the slope, which is balanced by the normal force. Understanding these components is essential for analyzing the skier's motion and determining how the angle of the slope affects the forces at play, leading to the relationship between the slope angle and the coefficient of kinetic friction.