A 1.5 -lb ball that can slide on a horizontal frictionless...

Key Concepts

Conservation of Angular Momentum

For a particle moving under a central force, its angular momentum remains constant if no external torques are acting on the system. This principle relates the tangential velocity to the radial distance at different stages of motion and is typically used when analyzing turning points in orbits or when the initial velocity is perpendicular to the radial direction.

Centripetal Force and Circular Motion

When an object moves perpendicular to the radius vector in a central force field, it undergoes circular or near?circular motion. The required centripetal force to maintain this motion is provided by the elastic force in the cord when it is stretched. This interplay between elastic restoring force and the need for centripetal acceleration is fundamental in analyzing the motion and determining minimum speeds or closest approaches.

Constraints on Tension in an Elastic Cord

In systems involving an elastic cord or spring attached to a moving object, ensuring that the cord remains taut requires that the tension (or restoring force provided by the stretched cord) stays non-negative. This condition imposes restrictions on the object's motion such that, for instance, the centripetal force required must not exceed the available elastic force, thereby preventing the cord from going slack.

Hooke's Law and Elastic Potential Energy

Hooke’s Law describes how the force exerted by an elastic element (or a spring) is proportional to its extension or compression from its natural length. The corresponding elastic potential energy is stored in the cord according to the formula (1/2)kx². Understanding this relationship is crucial when analyzing the energy interactions and restoring forces in systems with elastic constraints.

Conservation of Mechanical Energy

In systems without non?conservative forces, the total mechanical energy—comprising kinetic and potential energy—remains constant. In problems involving elastic elements, this means that the initial kinetic energy, along with any stored elastic potential energy, will convert between forms as the system evolves. This principle allows one to determine speeds, deformations, and distances by equating energy at different points in the motion.

Transcript

00:01 This is kind of an interesting problem.

00:03 It took me a little while to figure out what was exactly going on here.

00:07 So we have this ball, i'm attached to an elastic band here.

00:14 The elastic band has an unstretched length of x1.

00:19 X1 is, i think we're given it is two feet.

00:25 I guess i didn't write that down here, so i should put that in.

00:29 X1 equals two feet.

00:31 And x1 or x2 equals 1 foot so it's stretched total length initially is 3 feet and so then it's given some initial velocity around this point so it's basically swinging around here and now because the rubber band is stretched or the elastic band is stretched it's going to also want to come in and then also then go out and so in the end we should have kind of an oscillation thing about this about the center here since there's no friction now we're told that the weight of this thing is one and a half pounds the spring constant here is one pound per inch again i guess the initial length is right here is two feet then i guess i also have l1 here is three feet so that's the the form length is two feet the initial length is three feet initially we just have velocity in the horizontal direction or in the circumferential direction.

01:44 I should probably say that that's a little more accurate in general.

01:51 It just happens to be in the horizontal direction at this angle.

02:01 So we know we have then this velocity and again i'm put a two there, why did i put a two there? i'm not sure.

02:21 Anyway, we know what we're asked for then is what the the smallest low value of the initial speed if the chord is not to become slack.

02:40 So we weren't looking for this guy v0 2 whatever reason i put a 2 there so we're looking for that and we know that the chord will become slack if l the length at the present time is less than equal to 2.

03:07 So that would be a minimum distance.

03:09 So what we have here is we have some initial kinetic energy, this term, and some initial potential energy because this thing is stretched.

03:17 Now, in the case, when this thing is at a minimum distance so that it's just about to, you know, un, make the cord slack, we know that that distance, if that's going to be a minimum, then vr2 is zero.

03:40 So that's the one that took me a little while to figure out that we have this constraint on here because if we're cord is going to become.

03:48 Not to become slack, we need the minimum distance to be greater than l0.

03:56 So we have that and then we have our potential energy at this point, at the point where the distance is the minimum, or is l0.

04:06 So that's zero since l2 is l0.

04:10 And so what's troubling me at first is i didn't realize vr2 was zero, that again that this would be a minimum radius.

04:18 So i had too many unknowns here.

04:21 But if we get rid of this, then we can also use conservation of angular momentum because all the forces acting on this body go through this point.

04:33 So angle momentum is conserved.

04:34 So we can use conservation of angular momentum...

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