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 15b
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.
Verified step by step guidance1
Step 1: Recall the formula for the moment of inertia about a given axis: \( I = \sum m_i r_i^2 \), where \( m_i \) is the mass of each object and \( r_i \) is the perpendicular distance of the mass from the axis of rotation.
Step 2: Identify the axis of rotation. In this case, the axis passes through mass A (mass 1, \( 0.35 \, \text{kg} \)) and is perpendicular to the page. Therefore, the distance \( r \) for mass 1 is \( 0 \, \text{m} \) since it lies on the axis.
Step 3: Determine the distances of the other masses (mass 2 and mass 3) from the axis. For mass 2 (\( 0.35 \, \text{kg} \)), the distance is \( 0.30 \, \text{m} \). For mass 3 (\( 0.52 \, \text{kg} \)), the distance is \( 0.25 \, \text{m} \).
Step 4: Substitute the values into the formula for the moment of inertia. For mass 1, \( m_1 r_1^2 = 0.35 \times 0^2 \). For mass 2, \( m_2 r_2^2 = 0.35 \times (0.30)^2 \). For mass 3, \( m_3 r_3^2 = 0.52 \times (0.25)^2 \).
Step 5: Add the contributions from all masses to find the total moment of inertia: \( I = m_1 r_1^2 + m_2 r_2^2 + m_3 r_3^2 \). Simplify the expression to get the final result.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Moment of Inertia
The moment of inertia is a measure of an object's resistance to rotational motion about a specific axis. It depends on the mass distribution relative to that axis, calculated as the sum of the products of each mass and the square of its distance from the axis. For point masses, the formula is I = Σ(m_i * r_i^2), where m_i is the mass and r_i is the distance from the axis.
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Intro to Moment of Inertia
Axis of Rotation
The axis of rotation is an imaginary line around which an object rotates. In this problem, the axis passes through mass A and is perpendicular to the page, meaning that the moment of inertia must account for the distances of the other masses from this axis. Understanding the position of the axis is crucial for correctly calculating the moment of inertia.
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Mass Distribution
Mass distribution refers to how mass is spread out in an object or system. In this case, the three masses are connected by rigid rods, and their positions relative to the axis of rotation affect the overall moment of inertia. Analyzing the distances and weights of each mass helps in determining how they contribute to the system's resistance to rotation.
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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
The four masses shown in FIGURE EX12.13 are connected by massless, rigid rods. Find the coordinates of the center of mass.
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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 its center?
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Textbook Question
The three masses shown in FIGURE EX12.15 are connected by massless, rigid rods. Find the coordinates of the center of mass.
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Textbook Question
The four masses shown in FIGURE EX12.13 are connected by massless, rigid rods. Find the moment of inertia about a diagonal axis that passes through masses B and D.
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