Your roommate is working on his bicycle and has the bike upside down. He spins the 60-cm-diameter wheel, and you notice that a pebble stuck in the tread goes by three times every second. What are the pebble's speed and acceleration?

검증된 단계별 안내

Determine the radius of the wheel. The diameter is given as 60 cm, so the radius \( r \) is half of the diameter: \( r = \frac{60}{2} = 30 \, \text{cm} = 0.30 \, \text{m} \).

Calculate the angular velocity \( \omega \) of the wheel. The pebble completes 3 revolutions per second, so the angular velocity is \( \omega = 2 \pi \times 3 = 6 \pi \, \text{rad/s} \).

Find the pebble's linear speed \( v \). The relationship between linear speed and angular velocity is \( v = r \omega \). Substitute \( r = 0.30 \, \text{m} \) and \( \omega = 6 \pi \): \( v = 0.30 \times 6 \pi \).

Determine the pebble's centripetal acceleration \( a_c \). The formula for centripetal acceleration is \( a_c = \frac{v^2}{r} \). Substitute \( v = 0.30 \times 6 \pi \) and \( r = 0.30 \): \( a_c = \frac{(0.30 \times 6 \pi)^2}{0.30} \).

Simplify the expressions for \( v \) and \( a_c \) to find the pebble's speed and acceleration. Ensure the units are consistent and the results are expressed in \( \text{m/s} \) for speed and \( \text{m/s}^2 \) for acceleration.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Linear Speed

Linear speed refers to the distance traveled by an object per unit of time. In this scenario, the pebble's speed can be calculated by determining the circumference of the bicycle wheel and multiplying it by the number of revolutions per second. The formula for linear speed (v) is v = distance/time, where the distance is the circumference of the wheel.

Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circle. It can be calculated using the formula a_c = v²/r, where v is the linear speed and r is the radius of the circular path. This concept is crucial for understanding how the pebble accelerates as it moves along the wheel's circular trajectory.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a central point, expressed in radians per second. It is related to the number of revolutions per second and can be calculated using the formula ω = 2πf, where f is the frequency of rotation. In this case, knowing the angular velocity helps in determining both the linear speed and the centripetal acceleration of the pebble.