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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: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 12: Rotation of a Rigid BodyProblema 73b
Capitolo 12, Problema 73b

A long, thin rod of mass M and length L is standing straight up on a table. Its lower end rotates on a frictionless pivot. A very slight push causes the rod to fall over. As it hits the table, what are the speed of the tip of the rod?

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Start by identifying the type of motion involved. The rod rotates about the pivot point as it falls, so this is a rotational dynamics problem. The key is to use the principle of conservation of energy to relate the potential energy of the rod to its rotational kinetic energy as it falls.
Write the expression for the initial potential energy of the rod. The center of mass of the rod is located at a height of \( \frac{L}{2} \) above the table. The potential energy is given by \( U = M g \frac{L}{2} \), where \( M \) is the mass of the rod, \( g \) is the acceleration due to gravity, and \( L \) is the length of the rod.
As the rod falls, its potential energy is converted into rotational kinetic energy. The rotational kinetic energy is given by \( K = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia of the rod about the pivot point, and \( \omega \) is the angular velocity of the rod just before it hits the table. For a rod pivoting about one end, the moment of inertia is \( I = \frac{1}{3} M L^2 \).
Apply the conservation of energy principle: \( U_{initial} = K_{final} \). Substituting the expressions for potential energy and rotational kinetic energy, we get \( M g \frac{L}{2} = \frac{1}{2} \left( \frac{1}{3} M L^2 \right) \omega^2 \). Simplify this equation to solve for \( \omega \), the angular velocity of the rod just before it hits the table.
Finally, calculate the speed of the tip of the rod. The tip of the rod is at a distance \( L \) from the pivot point, so its linear speed is related to the angular velocity by \( v_{tip} = \omega L \). Substitute the value of \( \omega \) obtained in the previous step into this equation to find \( v_{tip} \).

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Rotational Motion

Rotational motion refers to the movement of an object around a central point or axis. In this scenario, the rod rotates about a pivot point at its lower end. Understanding how rotational motion works, including concepts like angular velocity and acceleration, is crucial for analyzing the dynamics of the falling rod.
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Equations of Rotational Motion

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. As the rod falls, its potential energy is converted into kinetic energy. This concept is essential for calculating the speed of the tip of the rod at the moment it strikes the table.
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Conservation Of Mechanical Energy

Kinematics of Rigid Bodies

Kinematics of rigid bodies involves the study of the motion of solid objects without considering the forces that cause the motion. In this case, analyzing the motion of the rod as it falls and the relationship between its angular displacement and the linear speed of its tip is key to determining the speed at impact.
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Free-Body Diagrams
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