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

Knight Calc 5th Edition

Ch 12: Rotation of a Rigid Body

Problem 71b

Chapter 12, Problem 71b

The 5.0 kg, 60-cm-diameter disk in FIGURE P12.71 rotates on an axle passing through one edge. The axle is parallel to the floor. The cylinder is held with the center of mass at the same height as the axle, then released. What is the cylinder's angular velocity when it is directly below the axle?

Verified step by step guidance

Step 1: Identify the physical principles involved. This problem involves rotational motion and energy conservation. The disk rotates about an axle passing through one edge, and its center of mass falls due to gravity. The potential energy lost by the center of mass is converted into rotational kinetic energy.

Step 2: Calculate the moment of inertia of the disk about the axle. Use the parallel axis theorem: \( I = I_{\text{center}} + m d^2 \), where \( I_{\text{center}} \) is the moment of inertia about the center of mass (\( \frac{1}{2} m r^2 \) for a disk), \( m \) is the mass of the disk, \( r \) is the radius, and \( d \) is the distance from the center of mass to the axle (equal to the radius in this case).

Step 3: Write the energy conservation equation. The initial potential energy of the disk's center of mass is \( U = m g h \), where \( h \) is the vertical distance the center of mass falls (equal to the radius of the disk). The final rotational kinetic energy is \( K = \frac{1}{2} I \omega^2 \), where \( \omega \) is the angular velocity.

Step 4: Solve for \( \omega \) using energy conservation. Set \( U = K \), substitute \( m g h \) for potential energy and \( \frac{1}{2} I \omega^2 \) for kinetic energy, and rearrange to find \( \omega \).

Step 5: Substitute the known values into the equations. Use \( m = 5.0 \ \text{kg} \), \( r = 0.30 \ \text{m} \), \( g = 9.8 \ \text{m/s}^2 \), and calculate \( I \) and \( \omega \) based on the derived formulas.

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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 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. Understanding this concept is crucial for analyzing the rotational dynamics of the disk in the problem.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the potential energy of the disk when held at a height is converted into kinetic energy as it falls and rotates. This relationship allows us to calculate the angular velocity of the disk when it reaches the lowest point.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second. It is related to linear velocity through the radius of the rotation, given by the formula ω = v/r, where ω is angular velocity, v is linear velocity, and r is the radius. In this problem, determining the angular velocity of the disk when it is directly below the axle is essential for understanding its motion.

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