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 73a

Chapter 4, Problem 73a

The angular velocity of a process control motor is ω = ( 20 - ½ t² ) rad/s, where t is in seconds. At what time does the motor reverse direction?

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Identify the condition for the motor to reverse direction. The motor reverses direction when its angular velocity (ω) becomes zero. This is because a change in direction implies that the velocity transitions through zero.

Set the given angular velocity equation to zero: ω = 20 - (1/2)t² = 0. Solve for t. Rearrange the equation to isolate t²: (1/2)t² = 20.

Multiply both sides of the equation by 2 to eliminate the fraction: t² = 40.

Take the square root of both sides to solve for t. Remember to consider both the positive and negative roots: t = ±√40. However, since time cannot be negative, only the positive root is physically meaningful.

Conclude that the motor reverses direction at t = √40 seconds. This is the time when the angular velocity becomes zero and the motor changes its rotational direction.

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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, expressed in radians per second. In this context, it is given as a function of time, indicating that the motor's speed changes as time progresses. Understanding angular velocity is crucial for analyzing rotational motion and determining when the direction of rotation changes.

Reversal of Direction

A motor reverses direction when its angular velocity changes from positive to negative. This occurs when the angular velocity function equals zero. By solving the equation for angular velocity, we can find the specific time at which this transition happens, indicating the moment the motor stops and starts rotating in the opposite direction.

Quadratic Functions

The angular velocity function provided is a quadratic equation in terms of time, represented as ω(t) = 20 - ½ t². Quadratic functions have a parabolic shape and can have one or two real roots, which correspond to the times when the function equals zero. Analyzing the roots of this quadratic equation is essential for determining the time at which the motor reverses direction.

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Textbook Question

A 25 g steel ball is attached to the top of a 24-cm-diameter vertical wheel. Starting from rest, the wheel accelerates at 470 rad/s². The ball is released after ¾ of a revolution. How high does it go above the center of the wheel?

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Textbook Question

A 6.0-cm-diameter gear rotates with angular velocity ω = ( 20 ─ ½ t² ) rad/s where t is in seconds. At t = 4.0 s, what are: The tangential acceleration of a tooth on the gear?

The angular velocity of a process control motor is ω = ( 20 ─ ½ t² ) rad/s, where t is in seconds. Through what angle does the motor turn between t = 0 s and the instant at which it reverses direction?

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Textbook Question

A 6.0-cm-diameter gear rotates with angular velocity ω = ( 20 ─ ½ t² ) rad/s where t is in seconds. At t = 4.0 s, what are: The gear's angular acceleration?

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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