Suppose the solid cylinder in the apparatus described in...

Suppose the solid cylinder in the apparatus described in Example 9.9 (Section 9.4$)$ is replaced by a thin-walled, hollow cylinder with the same mass $M$ and radius $R$ . The cylinder is attached to the axle by spokes of a negligible moment of inertia. (a) Find the speed of the hanging mass $m$ just as it strikes the floor. (b) Use energy concepts to explain why the answer to part (a) is different from the speed found in Example $9.9 .$

Key Concepts

Moment of Inertia

This concept quantifies how a body's mass is distributed with respect to the axis of rotation. It critically determines how much torque is needed for a desired angular acceleration. In rotating systems like cylinders, different mass distributions (e.g., solid versus thin-walled) lead to different moments of inertia, which in turn affect how the system responds when energy is added, altering the motion characteristics.

Rotational Kinetic Energy

Rotational kinetic energy is the energy a body possesses due to its rotation and is given by (1/2) I ?², where I is the moment of inertia and ? is the angular velocity. In systems where an object both rotates and translates, a part of the energy input (for example, from gravitational potential energy) is partitioned into rotational motion, thus affecting the overall speed of the system.

Conservation of Energy

The principle of conservation of energy states that the total energy in an isolated system remains constant over time. In the context of rotating systems, gravitational potential energy is converted into both translational and rotational kinetic energies. This principle helps explain differences in outcomes when the moment of inertia changes, as more energy stored in rotation leaves less available for translational motion.

Interplay Between Translational and Rotational Motion

In systems that undergo both translation and rotation, the distribution of energy between these modes dictates the system's behavior. A larger moment of inertia means more energy is channeled into rotational kinetic energy, which can result in a lower translational kinetic energy (and hence speed) compared to a system with a smaller moment of inertia. This interplay is crucial when comparing objects like solid and hollow cylinders under similar conditions.

Transcript

00:01 Hi, everybody.

00:01 So for this one, we need to find a, the speed of hanging mass.

00:07 Okay.

00:08 And as it strikes the floor, and that's what this second image right here is showing.

00:14 Okay.

00:16 You see how this mass? so what i'm going to call is we're going to call this part one.

00:21 And actually, let's do that in a different color.

00:25 Let's do that in green.

00:27 So we got one and we got two.

00:31 So for this, for this first one, which we have as one here, okay? we know that the mass is just going to focus on mge, okay? and that is because it's hanging with gravity and your kinetic energy initial is going to be zero.

00:54 Okay, because it's not moving.

00:55 And for potential energy of two and so what we can do is potential energy of two is just going to be zero because it's flying down okay and your kinetic energy of two is going to equal to one half mass times velocity squared and that's this one right here and your one plus one half the inertia and the velocity squared, your rotational velocity, which is this one right here, okay? because you have this wheel that's moving.

01:34 And now we can keep going to just make it look nicer.

01:38 We have nv squared plus one half.

01:42 Substitute capital m, r, okay, r squared times velocity squared.

01:51 Squared, okay? and what we can also do is sub in here for the rotational velocity divided by r squared, okay? and we can keep going.

02:07 And what i'm going to do is we can cross these bad boys out.

02:14 Perfect.

02:16 And now we're going to use a law of conservation.

02:19 So law conservation, we have k1.

02:21 Plus kinetic energy plus potential energy equals kinetic energy, second system, second one, okay? and so here we have zero plus mgh equals one half mb squared plus one half capital mr...

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