Textbook Question
A small 300 g ball and a small 600 g ball are connected by a 40-cm-long, 200 g rigid rod. b. What is the rotational kinetic energy if the structure rotates about its center of mass at 100 rpm?
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 55
Calculate the moment of inertia of the rectangular plate in FIGURE P12.55 for rotation about a perpendicular axis through the center.
Verified step by step guidance1
Step 1: Identify the formula for the moment of inertia of a rectangular plate about an axis perpendicular to its center. The formula is \( I = \frac{1}{12} M (L^2 + L^2) \), where \( M \) is the mass of the plate and \( L \) is the length of each side.
Step 2: Recognize that the plate is square, meaning both sides are equal in length (\( L \)). Substitute \( L \) for both dimensions in the formula.
Step 3: Simplify the formula by combining the terms \( L^2 + L^2 \) into \( 2L^2 \). The formula becomes \( I = \frac{1}{12} M (2L^2) \).
Step 4: Factor out the constant \( 2 \) to further simplify the formula to \( I = \frac{1}{6} M L^2 \).
Step 5: To calculate the moment of inertia, you would need the values of \( M \) (mass of the plate) and \( L \) (length of the sides). Plug these values into the formula \( I = \frac{1}{6} M L^2 \) to find the 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 the axis of rotation. For a rectangular plate, the moment of inertia can be calculated using the formula I = (1/12) * m * (L^2 + L^2), where m is the mass and L is the length of the sides.
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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 case, the axis is perpendicular to the plane of the rectangular plate and passes through its center. The choice of the axis significantly affects the moment of inertia, as it determines how the mass is distributed relative to that axis.
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Mass Distribution
Mass distribution refers to how mass is spread out in an object. For calculating the moment of inertia, it is crucial to know how the mass is arranged in relation to the axis of rotation. In a rectangular plate, the uniform distribution of mass affects the calculations, as each infinitesimal mass element contributes differently to the overall moment of inertia.
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Related Practice
Textbook Question
A person's center of mass is easily found by having the person lie on a reaction board. A horizontal, 2.5-m-long, 6.1 kg reaction board is supported only at the ends, with one end resting on a scale and the other on a pivot. A 60 kg woman lies on the reaction board with her feet over the pivot. The scale reads 25 kg. What is the distance from the woman's feet to her center of mass?
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Textbook Question
A 4.0-cm-diameter disk with a 3.0-cm-diameter hole rolls down a 50-cm-long, 20° ramp. What is its speed at the bottom? What percent is this of the speed of a particle sliding down a frictionless ramp?
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Textbook Question
Calculate by direct integration the moment of inertia for a thin rod of mass M and length L about an axis located distance d from one end. Confirm that your answer agrees with Table 12.2 when d=0 and when d = L/2.
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Textbook Question
Determine the moment of inertia about the axis of the object shown in FIGURE P12.52.
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Textbook Question
A 3.0-m-long ladder, as shown in Figure 12.35, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.40. What is the minimum angle the ladder can make with the floor without slipping?