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Ch 12: Rotation of a Rigid Body
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 12: Rotation of a Rigid BodyProblem 68
Chapter 12, Problem 68

Your engineering team has been assigned the task of measuring the properties of a new jet-engine turbine. You've previously determined that the turbine's moment of inertia is 2.6 kg m2. The next job is to measure the frictional torque of the bearings. Your plan is to run the turbine up to a predetermined rotation speed, cut the power, and time how long it takes the turbine to reduce its rotation speed by 50%. Your data are given in the table. Draw an appropriate graph of the data and, from the slope of the best-fit line, determine the frictional torque.

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Step 1: Understand the problem. The goal is to determine the frictional torque of the turbine bearings. The moment of inertia (I) is given as 2.6 kg·m², and the rotational speed decreases over time due to friction. The relationship between torque (τ), angular acceleration (α), and moment of inertia is given by the equation: τ=Iα.
Step 2: Analyze the data. The data table provides the rotational speed (ω) at different times. To find the angular acceleration (α), calculate the rate of change of angular velocity with respect to time. Use the formula: α=ΔωΔt, where Δω is the change in angular velocity and Δt is the change in time.
Step 3: Plot the data. Create a graph of angular velocity (ω) versus time (t). The slope of the best-fit line on this graph represents the angular acceleration (α). Ensure the graph is properly labeled with units on both axes.
Step 4: Determine the slope of the best-fit line. Use the data points to calculate the slope of the line, which corresponds to the angular acceleration (α). The slope can be calculated using the formula: α=ω2−ω1t2−t1, where ω1 and ω2 are angular velocities at times t1 and t2, respectively.
Step 5: Calculate the frictional torque. Once the angular acceleration (α) is determined from the slope, use the equation τ=Iα to calculate the frictional torque. Substitute the given moment of inertia (I = 2.6 kg·m²) and the calculated angular acceleration (α) into the equation to find the torque.

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Key Concepts

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

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution of the object relative to the axis of rotation. In this case, the turbine's moment of inertia is given as 2.6 kg m², indicating how difficult it is to change its rotational speed. A higher moment of inertia means more torque is required to achieve the same angular acceleration.
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Frictional Torque

Frictional torque is the torque that opposes the motion of a rotating object due to friction in its bearings or other components. It can be calculated by analyzing the deceleration of the object when external power is removed. In this scenario, measuring how long it takes for the turbine to slow down by 50% allows for the determination of the frictional torque acting on it, which is crucial for understanding the efficiency and performance of the turbine.
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Graphing and Linear Regression

Graphing the data collected from the turbine's deceleration allows for visual analysis of the relationship between time and angular velocity. By plotting the data points and applying linear regression, one can find the best-fit line that represents the trend. The slope of this line is directly related to the frictional torque, providing a quantitative measure of how quickly the turbine slows down, which is essential for evaluating its performance.
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Related Practice
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