Ch 09: Work and Kinetic Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 09: Work and Kinetic Energy

문제 49b

9장, 문제 49b

A 50 kg ice skater is gliding along the ice, heading due north at 4.0 m/s. The ice has a small coefficient of static friction, to prevent the skater from slipping sideways, but μ_k = 0. Suddenly, a wind from the northeast exerts a force of 4.0 N on the skater. What is the minimum value of μ_s that allows her to continue moving straight north?

검증된 단계별 안내

Step 1: Identify the forces acting on the skater. The skater is moving due north, and a wind from the northeast exerts a force of 4.0 N. The ice provides a static friction force to counteract the sideways component of the wind force. Since μk = 0, the kinetic friction is negligible, and only static friction is relevant.

Step 2: Break the wind force into components. The wind is coming from the northeast, which means it has both an eastward and northward component. Use trigonometry to find the components of the wind force: \( F_{east} = F \cdot \cos(45^\circ) \) and \( F_{north} = F \cdot \sin(45^\circ) \), where \( F = 4.0 \, \text{N} \).

Step 3: Determine the role of static friction. The static friction force must counteract the eastward component of the wind force to prevent the skater from slipping sideways. The maximum static friction force is given by \( F_{friction} = \mu_s \cdot F_{normal} \), where \( F_{normal} \) is the normal force exerted by the ice. For the skater, \( F_{normal} = m \cdot g \), where \( m = 50 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \).

Step 4: Set up the inequality for static friction. To ensure the skater continues moving straight north, the static friction force must be at least equal to the eastward component of the wind force: \( \mu_s \cdot F_{normal} \geq F_{east} \). Substitute \( F_{normal} = m \cdot g \) and \( F_{east} = F \cdot \cos(45^\circ) \) into the inequality.

Step 5: Solve for \( \mu_s \). Rearrange the inequality to isolate \( \mu_s \): \( \mu_s \geq \frac{F \cdot \cos(45^\circ)}{m \cdot g} \). Substitute the known values for \( F \), \( m \), and \( g \) to find the minimum value of \( \mu_s \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

주요 개념

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

Friction

Friction is the force that opposes the relative motion of two surfaces in contact. It is characterized by two coefficients: static friction (μs), which prevents motion, and kinetic friction (μk), which acts when surfaces are sliding against each other. In this scenario, static friction is crucial as it must be sufficient to counteract the lateral force exerted by the wind, allowing the skater to maintain a straight path.

Force and Motion

Newton's laws of motion describe the relationship between the forces acting on an object and its motion. The first law states that an object at rest or in uniform motion will remain so unless acted upon by a net external force. In this case, the wind exerts a force that could change the skater's direction, necessitating a balance of forces to keep her moving straight north.

Equilibrium

Equilibrium occurs when the net force acting on an object is zero, resulting in no change in motion. For the skater to continue moving straight north despite the wind's force, the static friction must equal the wind's force. This balance ensures that the skater does not slip sideways, highlighting the importance of calculating the minimum static friction coefficient required for equilibrium.

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