A car tire is 60 cm in diameter. The car is traveling at a speed of 20 m/s. What is the speed of a point at the top edge of the tire?

검증된 단계별 안내

Step 1: Understand the problem. The car tire is rotating while the car is moving forward. The speed of a point at the top edge of the tire is the sum of the linear speed of the car and the tangential speed of the tire due to its rotation.

Step 2: Calculate the radius of the tire. The diameter is given as 60 cm, so the radius \( r \) is half of the diameter: \( r = \frac{60}{2} = 30 \, \text{cm} = 0.3 \, \text{m} \).

Step 3: Determine the angular velocity \( \omega \) of the tire. The linear speed of the car \( v \) is related to the angular velocity by the formula \( v = r \omega \). Rearrange to find \( \omega = \frac{v}{r} \). Substitute \( v = 20 \; \text{m/s} \) and \( r = 0.3 \; \text{m} \).

Step 4: Calculate the tangential speed of a point on the tire due to its rotation. The tangential speed at the top edge of the tire is equal to \( r \omega \), which is the same as the linear speed of the car, \( v = 20 \; \text{m/s} \).

Step 5: Add the linear speed of the car and the tangential speed of the tire at the top edge. The total speed of a point at the top edge of the tire is \( v_{\text{top}} = v + v = 20 + 20 = 40 \; \text{m/s} \).

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주요 개념

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

Linear Speed

Linear speed refers to the distance traveled per unit of time. In this context, it is the speed at which the car is moving, which is given as 20 m/s. This speed is crucial for understanding how the motion of the car translates to the motion of points on the tire.

Rotational Motion

Rotational motion describes how an object rotates around an axis. For a tire, this involves the relationship between its linear speed and the angular speed, which is determined by the radius of the tire. Understanding this concept helps in calculating the speed of points on the tire as it rolls.

Tangential Speed

Tangential speed is the linear speed of a point on the circumference of a rotating object. At the top edge of the tire, the tangential speed is the sum of the car's linear speed and the speed due to the tire's rotation. This concept is essential for determining the speed of a point at the top edge of the tire.