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A $20.00-\mathrm{kg}$ lead sphere is hanging from a hook by a thin wire 3.50 $\mathrm{m}$ long, and is free to swing in a complete circle. Suddenly it is struck horizontally by a $5.00-\mathrm{kg}$ steel dart that embeds itself in the lead sphere. What must be the minimum initial speed of the dart so that the combination makes a complete circular loop after the collision?

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$v_{1}=65.5 \mathrm{m} / \mathrm{s}$

Physics 101 Mechanics

Chapter 8

Momentum, Impulse, and Collisions

Moment, Impulse, and Collisions

Cornell University

University of Michigan - Ann Arbor

Hope College

Lectures

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

03:30

In physics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Given a force, F, applied for a time, t, the resulting change in momentum, p, is equal to the impulse, I. Impulse applied to a mass, m, is also equal to the change in the object's kinetic energy, T, as a result of the force acting on it.

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here in discussion. The given figure represents the forces acting on the combined system at the top most point popped up. Now we can express the relation between the forces acting on the combined system at the top most point of the circle as fc. Equal to E Plus W. Now we considered it as equation what? Now? Here fc is thus a contributor force acting on the combined system to keep it in a circular path. T. Is the tension in the string and W. Is the weight of the combined system. Now we know that expression for the centripetal force can be lit. Tennis fc equals two. Mm. I'm buying we square. Oh yeah, I did buy are now we consider it as equation you. So here Fc is the centripetal force acting on the combined system. Mm kong Is the mask of the combined system we got is the speed of the combined system at the top of the second. Now, similarly the expression for the vehicle, the combined system can be read and health W. Equal to and combined, multiplied by the. Now we consider it as equation. See here, W represents the weight of the combined system and come. Is that mars of the combined strategy. Is the isolation due to good idea now. Yeah. Really right equation one help of equation to an equation. Mhm. So yeah, we get and I'm buying what people are about to be square dog divided by our It goes to he Yes. Um A mine were deep lava Z. Now we considered it has equation for No. For the calculation of the minimum spirit required for the combined system to make a complete circular group attention. Industrial P must be quite zero. That means here he must be equal to zero mm. They sub studio The equal to zero and the equation for therefore we get um I'm buying multiple ever. These are no, you are guided by art Equal to zero plus. um I'm fine marty played G. Now here, that's really like this for we squared off. Therefore we get we square oh equals true G. Market. Uh So from about we get equals two the square root of G. Multi level are therefore the speed of the combined system at the top of the circle is square loop of jean multiply by mhm No. Yeah, We substitute 3.50 m for RN 9.8% in the square for G. So we get we talk equal to square root off 9.8 multiply by 3.50. So we get the oh it wants to five point 856 meter or second. Therefore the speed of the combined system at the top of the circle is 5.856 m/s. No, we apply principle of conservation of energy. To find the high speed of the combined system. At that boat ramp up to circle. Oh, it means that eagle must be equal to is border. Now here it opens the total energy of the combined system at the top of the circling each canister, total energy of the combined system of the bottom of the circuit. Now at the bottom of the circle, the combined system possessions only that planet energy and it is due to the embedding of the movie is still dark with that lead this fear. So at the top of the circle, that combined system for the society energy act gravitational potential energy. So here We use one divided by two. And for my people have a v bottom square for the bottom and one divided by two. And from the heat of the square plus and congenial people have a two or four ito to find the border. So we can like this as one divided by two. And and are you glad to be scrapped? Yes. And um, But people have a team where people have a two part equal to but well I did buy and um my deep love right we bored don't one is square. So from a bow we get we bought them. It will stop The square root of five million people are 100 clarity. So this is the speed of the combined system at the bottom of to Syracuse. Now here the sub studio mm on 50 for us And 9.84 G. Therefore we get to be a border town it close to the square root of five while people are about 3.50 Married 11 9.8. So we get to the bottom it was you. 13.1 m but second. So the speed of the combined system at the bottom of the circle is 13.1. Make up for a second. No, on applying principle. Uh, an observation of momentum bigger mds. We won. It will do I am less mm gas the border now here and this is the most of the steel dart and m is the mass of the atmosphere. And we even is that initially speed of the steel? No, so now here the substitute five easy for em death 20 easy for em 13.1. Wait up a second fort we brought up to find weaver. So we get five multi lullaby. We one equals two. 20 plus why married people are by 13.1. So from mobile we get we one equals two 65.5 m. But second and the minimal initial spirit of the steel darts so that the combination makes a complete circular loop after coalition is 65.5 seconds

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