Textbook Question
A 25 kg solid door is 220 cm tall, 91 cm wide. What is the door's moment of inertia for rotation on its hinges?
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Ch 12: Rotation of a Rigid Body
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 12, Problem 17a
A 12-cm-diameter DVD has a mass of 21 g. What is the DVD’s moment of inertia for rotation about a perpendicular axis through its center?
Verified step by step guidance1
Step 1: Identify the shape of the DVD. The DVD can be approximated as a uniform solid disk. The formula for the moment of inertia of a solid disk rotating about an axis perpendicular to its center is \( I = \frac{1}{2} m r^2 \), where \( m \) is the mass and \( r \) is the radius.
Step 2: Convert the given diameter of the DVD into its radius. The diameter is 12 cm, so the radius \( r \) is half of the diameter: \( r = \frac{12}{2} = 6 \, \text{cm} \). Convert this to meters: \( r = 0.06 \, \text{m} \).
Step 3: Convert the mass of the DVD from grams to kilograms. The mass is given as 21 g, and since \( 1 \, \text{kg} = 1000 \, \text{g} \), the mass in kilograms is \( m = \frac{21}{1000} = 0.021 \, \text{kg} \).
Step 4: Substitute the values of \( m \) and \( r \) into the formula for the moment of inertia: \( I = \frac{1}{2} m r^2 \). This becomes \( I = \frac{1}{2} \times 0.021 \times (0.06)^2 \).
Step 5: Perform the calculation to find the moment of inertia. Ensure that the units are consistent (mass in kilograms and radius in meters) and simplify the expression to obtain the final value of \( I \).
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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 relative to the axis of rotation. For a solid disk, the moment of inertia can be calculated using the formula I = (1/2) m r², where m is the mass and r is the radius.
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Rotational Axis
The rotational axis is the line about which an object rotates. In this case, the axis is perpendicular to the plane of the DVD and passes through its center. The choice of axis significantly affects the moment of inertia, as it determines how the mass is distributed relative to that axis.
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Mass and Radius
The mass and radius of an object are critical parameters in calculating its moment of inertia. The mass (21 g) indicates how much matter is present, while the radius (6 cm, derived from the 12 cm diameter) affects how far the mass is distributed from the axis of rotation. Both factors are essential for determining the object's resistance to rotational acceleration.
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Related Practice
Textbook Question
A 25 kg solid door is 220 cm tall, 91 cm wide. What is the door’s moment of inertia for rotation about a vertical axis inside the door, 15 cm from one edge?
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Textbook Question
In FIGURE EX12.19, for what value of Xaxle will the two forces provide 1.8 Nm of torque about the axle?
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Textbook Question
The three masses shown in FIGURE EX12.15 are connected by massless, rigid rods. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page.
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Textbook Question
A 12-cm-diameter DVD has a mass of 21 g. What is the DVD's moment of inertia for rotation about a perpendicular axis through the edge of the disk?
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Textbook Question
A 4.0-m-long, 500 kg steel beam extends horizontally from the point where it has been bolted to the framework of a new building under construction. A 70 kg construction worker stands at the far end of the beam. What is the magnitude of the torque about the bolt due to the worker and the weight of the beam?
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