An experiment is performed in deep space with two uniform...

An experiment is performed in deep space with two uniform spheres, one with mass 50.0 $\mathrm{kg}$ and the other with mass 100.0 $\mathrm{kg} .$ They have equal radii, $r=0.20 \mathrm{m} .$ The spheres are released from rest with their centers 40.0 $\mathrm{m}$ apart. They accelerate toward each other because of their mutual gravitational attraction. You can ignore all gravitational forces other than that between the two spheres. (a) Explain why linear momentum is conserved. (b) When their centers are 20.0 m apart, find (i) the speed of each sphere and (ii) the magnitude of the relative velocity with which one sphere is approaching the other. (c) How far from the initial position of the center of the 50.0-kg sphere do the surfaces of the two spheres collide?

Key Concepts

Gravitational Attraction in Two-Body Systems

Gravitational force is a central force acting along the line joining two masses, proportional to the product of their masses and inversely proportional to the square of their separation distance. This interaction is fundamental in classical mechanics, especially in problems involving orbits, free-fall, and mutual acceleration. Recognizing the nature of this force allows one to apply energy methods and equations of motion effectively in isolated systems.

Collision Dynamics of Spherical Bodies

When dealing with colliding objects, especially spheres, it is important to understand the conditions under which they make contact. This involves knowing that the collision happens when the distance between the centers of the spheres equals the sum of their radii. This concept is critical for determining the position and timing of collisions and for analyzing the energy and momentum exchanges that occur during the impact.

Conservation of Energy

In mechanical systems, energy conservation implies that the total energy remains constant over time, provided that no non-conservative forces (such as friction) are doing work. In gravitational interactions, potential energy is converted into kinetic energy as bodies accelerate toward one another. This conversion allows one to compute changes in speed by equating the loss in gravitational potential energy with the gain in kinetic energy.

Conservation of Linear Momentum

This concept states that the total linear momentum of a system remains constant if no external forces are acting on it. In isolated systems, such as objects interacting in deep space where only internal forces (like gravitational force between two bodies) are present, the momentum lost by one object is exactly gained by the other. This principle is fundamental in analyzing collisions and interactions in mechanics, ensuring that the vector sum of momenta remains unchanged.

Center of Mass and Relative Motion

The concept of the center of mass is used to simplify the analysis of a system of interacting bodies by focusing on the weighted average position of all the mass in the system. In collisions or mutual gravitational attractions, analyzing the motion relative to the center of mass helps determine the individual velocities of the objects and the overall relative velocity between them, as it isolates the internal dynamics from any external influences.

Transcript

00:01 We're given the following situation with two spheres in space where no other forces can act on them except their own gravitation.

00:08 So we have m one of 50 kilograms and then 14 years away, m 2 100 kilograms were also given the radius is of the sphere.

00:15 But that's not gonna matter because they're never directly touching each other.

00:19 We can always just treat them each as point particles.

00:22 So, um, i'll just treat them like that here.

00:26 So first we're asked to explain in part a while in your momentum is conserved.

00:31 So you get into this.

00:32 A lot of reasons, um, make a fairly intuitive one is just that there are no external forces.

00:42 So the only forces acting are m one on em to an m two on m one.

00:47 So that means that's the total mental of the system.

00:50 Must be conserved, can't change soapy initial past equal p final eso part b then says when their centers or 20 years of parts we want to find the speed of each fear and then the magnitude, the relative velocity, um, of each fear.

01:10 So, um, that will be not when they've moved from 42 them both being a little bit closer to each other and now only 20 meters apart.

01:23 So we'll do this with conservation of energy.

01:25 So will say that the initial energy must equal the final energy of this system.

01:32 So we have an initial kinetic energy plus an initial potential energy.

01:37 And we have a final kinetic energy plus a final potential energy.

01:45 So ah, initially, the spheres are at rest.

01:47 So this initial connecticut and she is just zero.

01:50 So we only have our initial kinetic energy are sorry.

01:53 Our initial potential energy so negative g masses of the two spheres and then divided by their initial distance of 40 meters.

02:04 And then we have that equaling the final kinetic energy.

02:11 And then the final potential energy is so negative g and one and two and then the final distance of 20 meters.

02:20 So we can get then that the total kinetic energy in the end must be negative.

02:27 G m on them too, over 40 and then plus g m one m two over 20.

02:37 And so we can evaluate that and we get 8.34 times 10 to the minus nine.

02:45 So nano jules so that will then help us fair with the velocities of them.

02:51 Are so need thio actually rely on this fact from part a, that there's no external forces and so that the initial momentum must equal the final immense, um so because the initial momentum is zero, that means the final months must always be zero.

03:09 And so that means that these two masses have to have equal and opposites momentum's at all time so we can say assuming m one is traveling to the rights of deposit direction.

03:20 Then we'll have m one v one and then plus m to v to use of the momentum of each of them.

03:28 Has to be zero had any points, and so we can solve review to then and say v.

03:34 Two needs to be negative and one over him, too.

03:38 Times v one and so am ones...

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