Flywheels—rapidly rotating disks—are widely used in industry for storing energy. They are spun up slowly when extra energy is available, then decelerate quickly when needed to supply a boost of energy. A 20-cm-diameter rotor made of advanced materials can spin at 100,000 rpm. What is the speed of a point on the rim of this rotor?

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Convert the diameter of the rotor to the radius. The diameter is 20 cm, so the radius is half of that: \( r = \frac{20}{2} \) cm = 10 cm = 0.1 m (convert to meters for SI units).

Convert the rotational speed from revolutions per minute (rpm) to radians per second. Use the formula: \( \omega = \text{rpm} \times \frac{2\pi}{60} \). Substituting \( \text{rpm} = 100,000 \), we get \( \omega = 100,000 \times \frac{2\pi}{60} \) rad/s.

Use the formula for the tangential speed of a point on the rim of a rotating object: \( v = r \cdot \omega \), where \( r \) is the radius and \( \omega \) is the angular velocity in radians per second.

Substitute the values of \( r = 0.1 \) m and \( \omega \) (calculated in step 2) into the formula \( v = r \cdot \omega \).

Simplify the expression to find the tangential speed \( v \). Ensure the units are consistent (meters per second) and verify the calculation.

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주요 개념

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

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second. For a rotating object, it can be calculated from the rotational speed in revolutions per minute (rpm) by converting it to radians per second using the formula: angular velocity (ω) = (rpm × 2π) / 60.

Linear Speed

Linear speed refers to the distance traveled by a point on the circumference of a rotating object per unit of time. It can be calculated from the angular velocity and the radius of the object using the formula: linear speed (v) = angular velocity (ω) × radius (r). This relationship shows how the speed at the edge of a rotating disk increases with both the rotation rate and the size of the disk.

Radius of Rotation

The radius of rotation is the distance from the center of a rotating object to a point on its edge. In this case, the rotor has a diameter of 20 cm, which means its radius is 10 cm (or 0.1 m). The radius is crucial for calculating the linear speed of a point on the rim, as it directly influences how far that point travels in a given time based on the angular velocity.