Ch 04: Kinematics in Two Dimensions

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 04: Kinematics in Two Dimensions

Problem 71a

Chapter 4, Problem 71a

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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Textbook Question

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. b. Suppose the rotor's angular velocity decreases by 40% over 30 s as it supplies energy. What is the magnitude of the rotor's angular acceleration? Assume that the angular acceleration is constant.

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