Find the linear speed v for each of the following.

a point on the edge of a flywheel of radius 2 m, rotating 42 times per min

검증된 단계별 안내

Identify the given values: the radius of the flywheel \(r = 2\) meters, and the rotational speed \(f = 42\) revolutions per minute (rpm).

Recall the formula for linear speed \(v\) of a point on the edge of a rotating object: \(v = r \times \omega\), where \(\omega\) is the angular velocity in radians per unit time.

Convert the rotational speed from revolutions per minute to angular velocity in radians per second using the conversion: \(\omega = 2\pi \times \frac{f}{60}\), because one revolution equals \(2\pi\) radians and there are 60 seconds in a minute.

Substitute the values into the angular velocity formula: \(\omega = 2\pi \times \frac{42}{60}\) radians per second.

Finally, calculate the linear speed by multiplying the radius by the angular velocity: \(v = 2 \times \omega\). This will give the linear speed in meters per second.

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주요 개념

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Angular Velocity

Angular velocity measures how fast an object rotates, typically expressed in radians per second. It relates the number of rotations per unit time to the angle covered, where one full rotation equals 2π radians. Converting rotations per minute (rpm) to radians per second is essential for calculations involving rotational motion.

Linear Speed in Circular Motion

Linear speed (v) of a point on a rotating object is the distance traveled per unit time along the circular path. It is related to angular velocity (ω) by the formula v = ωr, where r is the radius. This relationship connects rotational motion to linear motion at the edge of the rotating body.

Unit Conversion and Application

Accurate unit conversion is crucial, especially converting rpm to radians per second and ensuring radius units are consistent. Applying the correct formula with proper units allows for precise calculation of linear speed, which is vital in practical problems involving rotating objects like flywheels.

Find the linear speed v for each of the following.a point on the edge of a flywheel of radius 2 m, rotating 42 times per min