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 73b

Chapter 4, Problem 73b

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?

Verified step by step guidance

Step 1: Understand the problem. The angular velocity ω is given as a function of time: ω = 20 - (1/2)t² rad/s. The motor reverses direction when ω = 0. To find the angle θ turned by the motor, we need to integrate the angular velocity over the time interval from t = 0 to the time when ω = 0.

Step 2: Determine the time at which the motor reverses direction. Set ω = 0 and solve for t: 0 = 20 - (1/2)t². Rearrange the equation to isolate t²: t² = 40. Solve for t: t = √40 seconds.

Step 3: Write the expression for the angle θ turned by the motor. The angle θ is the integral of angular velocity ω with respect to time: θ = ∫ω dt. Substitute ω = 20 - (1/2)t² into the integral: θ = ∫(20 - (1/2)t²) dt.

Step 4: Perform the integration. Break the integral into two parts: θ = ∫20 dt - ∫(1/2)t² dt. The first term integrates to 20t, and the second term integrates to -(1/6)t³. Combine these results: θ = 20t - (1/6)t³.

Step 5: Evaluate θ over the interval from t = 0 to t = √40. Substitute the limits of integration into the expression for θ: θ = [20(√40) - (1/6)(√40)³] - [20(0) - (1/6)(0)³]. Simplify the expression to find the total angle turned by the motor.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Angular Velocity

Angular velocity (ω) is a measure of how quickly an object rotates around an axis, expressed in radians per second. In this context, the angular velocity of the motor is given as a function of time, indicating that it changes as time progresses. Understanding angular velocity is crucial for determining how the motor's speed varies and for calculating the total angle turned over a specific time interval.

Angular Displacement

Angular displacement refers to the angle through which an object has rotated about a specific axis, measured in radians. To find the total angle the motor turns, we need to integrate the angular velocity function over the time interval from t = 0 s to the time when the motor reverses direction. This concept is essential for connecting the motor's changing speed to the total angle it covers.

Reversal of Direction

The reversal of direction in a rotating object occurs when its angular velocity changes sign, indicating a transition from clockwise to counterclockwise rotation or vice versa. In this problem, we need to determine the time at which the angular velocity becomes zero (ω = 0) to find when the motor reverses direction. This point is critical for calculating the angle turned during the specified time interval.

Recommended video:

09:49

Direction of Current in Loop Equations

Related Practice

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?

views

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?

views

Textbook Question

A painted tooth on a spinning gear has angular position θ = (6.0 rad/s⁴)t⁴. What is the tooth's angular acceleration at the end of 10 revolutions?

views

Textbook Question

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?

views

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.

views