A 500.0 g bird is flying horizontally at 2.25 m / s, not paying much attention, when it sudden

A 500.0 g bird is flying horizontally at 2.25 m / s, not...

A $500.0 \mathrm{~g}$ bird is flying horizontally at $2.25 \mathrm{~m} / \mathrm{s}$, not paying much attention, when it suddenly flies into a stationary vertical bar, hitting it $25.0 \mathrm{~cm}$ below the top (Fig. P10.85). The bar is uniform, $0.750 \mathrm{~m}$ long, has a mass of $1.50 \mathrm{~kg}$, and is hinged at its base. The collision stuns the bird so that it just drops to the ground afterward (but soon recovers to fly happily away).What is the angular velocity of the bar (a) just after it is hit by the bird and (b) just as it reaches the ground?

Key Concepts

Conservation of Mechanical Energy in Rotational Systems

When the bar swings down after being hit, gravitational potential energy is converted into rotational kinetic energy. By applying the conservation of mechanical energy, which states that the total mechanical energy remains constant in the absence of non-conservative forces, one can relate the loss in potential energy to the gain in rotational kinetic energy and thereby determine the angular velocity at various positions during the motion.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, depending on how its mass is distributed relative to the axis of rotation. For a uniform rod pivoted at one end, the moment of inertia is given by I = (1/3)mL². This concept is essential when relating the torque or impulse applied to an object to its angular velocity.

Conservation of Angular Momentum

In rotational dynamics, the conservation of angular momentum principle states that if no external torques act on a system, the total angular momentum remains constant. This concept is crucial when analyzing collisions in rotational systems, such as a bird striking a bar, where the bird's linear momentum is converted into angular momentum of the bar during a nearly instantaneous collision.

Transcript

00:01 And the angular momentum is conserved but mechanical and it is not conserved.

00:04 So direct collision during collision will be applying conservation off angular momentum.

00:13 And after the collision will be applying the energy conservation.

00:18 Since the mechanical energy is conserved, so after collision will be a playing conservation off energy.

00:31 Okay, so first of all, we should see here that when the bird hits the pole before hitting it ah, the bar, the bird will have an angler momentum off m bird times uh, the length off the pole, times velocity.

00:55 So these the linear velocity on da this distance from here to the hinge is 50 centimetres.

01:07 So we'll be using that.

01:10 So it's gonna be 0.5 meter so and ah, we're already given the mass and velocity so we can figure out that that and ah, also we should note about the we should take a notable the moment of inertia off the road, which is let's call it i on.

01:37 And for that it's gonna be 1/3 m bye and squared.

01:45 So from here, if we use the numbers embodies given us 1.5 kg and l is point 75 meters.

02:03 Ah, using that.

02:04 We see that moment off.

02:05 Initiates 0.78 sorry.

02:09 0.281 uh, the unit is k g meter squared and also here.

02:23 We need to take a note on the, uh, energy conservation.

02:28 So we see that when we're when we deal with conservation of energy, we take the center of mass of the bar.

02:36 Toby, uh, the center are toby to be the center where all the masses, uh, at so basically, if if that's the center of the bar, then we'll say that the entire mass is confined here.

02:56 And if we consider this as our, um initial point or if we said this as our zero reference point, then when the bar drops to the ground, um, this distance right here will be why equals negative three for a negative 0.375 years.

03:21 So, yeah, we'll be using that in in a second.

03:25 When will be applying conservation of energy will discuss more about it.

03:29 But let's first ah, do the angular momentum conservation...

Please give Ace some feedback