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Chapter 8: Problem 16

Can the center of mass of an object be located at a point outside the object, that is, at a point in space where no part of the object is located? Explain.

Short Answer

Expert verified
Explain. Answer: Yes, the center of mass of an object can be located at a point outside the object for non-homogeneous objects or composite systems. This may occur when the mass distribution is uneven or when there is empty space between individual objects in a system. Examples of such objects include boomerang-shaped or horseshoe-shaped objects, and composite systems like two spheres connected by a thin rod with different masses and distances between them.

Step by step solution

01

Understanding the Center of Mass

The center of mass is defined as the average location of the mass of an object. It is a point at which the object can be balanced perfectly under the influence of gravity.
02

Considering the Center of Mass for Homogeneous Objects

For regular, homogeneous objects (meaning objects with constant density throughout), the center of mass lies at the geometrical center. In this case, the center of mass will always be within the object. For example, the center of mass for a uniform sphere or a rectangular block will lie exactly at the geometrical center of these objects.
03

Investigating the Center of Mass for Non-homogeneous Objects

For non-homogeneous objects (objects with varying density throughout), the center of mass might not lie at the geometrical center. Depending on the mass distribution, it is possible that the center of mass can be located outside the object. For example, consider a boomerang-shaped object or a horseshoe-shaped object. Due to their shapes, their center of mass might not be at their geometrical center and can be located outside the object in the middle of the empty space of the shape.
04

Center of Mass in Composite Systems

Another instance where the center of mass can lie outside an object is in composite systems, which consist of multiple individual objects. In this case, the center of mass may lie in the empty space between the individual objects. For example, consider two spheres connected by a thin rod. If the masses and distances between the spheres are different, the center of mass of this composite system might not be inside any of the individual spheres or the connecting rod, but rather located somewhere in the empty space between them.
05

Conclusion

In conclusion, the center of mass can be located at a point outside the object for non-homogeneous objects or composite systems. This occurs when the mass distribution is uneven or when there is empty space between individual objects in a system. Understanding the concept of center of mass and its location is essential for analyzing the motion and stability of objects under the influence of external forces.

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Most popular questions from this chapter

Find the center of mass of a rectangular plate of length \(20.0 \mathrm{~cm}\) and width \(10.0 \mathrm{~cm} .\) The mass density varies linearly along the length. At one end, it is \(5.00 \mathrm{~g} / \mathrm{cm}^{2}\); at the other end, it is \(20.0 \mathrm{~g} / \mathrm{cm}^{2}\)

A system consists of two particles. Particle 1 with mass \(2.0 \mathrm{~kg}\) is located at \((2.0 \mathrm{~m}, 6.0 \mathrm{~m})\) and has a velocity of \((4.0 \mathrm{~m} / \mathrm{s}, 2.0 \mathrm{~m} / \mathrm{s}) .\) Particle 2 with mass \(3.0 \mathrm{~kg}\) is located at \((4.0 \mathrm{~m}, 1.0 \mathrm{~m})\) and has a velocity of \((0,4.0 \mathrm{~m} / \mathrm{s})\) a) Determine the position and the velocity of the center of mass of the system. b) Sketch the position and velocity vectors for the individual particles and for the center of mass.

Find the following center-of-mass information about objects in the Solar System. You can look up the necessary data on the Internet or in the tables in Chapter 12 of this book. Assume spherically symmetrical mass distributions for all objects under consideration. a) Determine the distance from the center of mass of the Earth-Moon system to the geometric center of Earth. b) Determine the distance from the center of mass of the Sun-Jupiter system to the geometric center of the Sun.

An artillery shell is moving on a parabolic trajectory when it explodes in midair. The shell shatters into a very large number of fragments. Which of the following statements is true (select all that apply)? a) The force of the explosion will increase the momentum of the system of fragments, and so the momentum of the shell is not conserved during the explosion. b) The force of the explosion is an internal force and thus cannot alter the total momentum of the system. c) The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the last fragment touches the ground. d) The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the first fragment touches the ground. e) The center of mass of the system of fragments will have a trajectory that depends on the number of fragments and their velocities right after the explosion.

The Saturn \(V\) rocket, which was used to launch the Apollo spacecraft on their way to the Moon, has an initial mass \(M_{0}=2.80 \cdot 10^{6} \mathrm{~kg}\) and a final mass \(M_{1}=8.00 \cdot 10^{5} \mathrm{~kg}\) and burns fuel at a constant rate for \(160 .\) s. The speed of the exhaust relative to the rocket is about \(v=2700 . \mathrm{m} / \mathrm{s}\). a) Find the upward acceleration of the rocket, as it lifts off the launch pad (while its mass is the initial mass). b) Find the upward acceleration of the rocket, just as it finishes burning its fuel (when its mass is the final mass). c) If the same rocket were fired in deep space, where there is negligible gravitational force, what would be the net change in the speed of the rocket during the time it was burning fuel?
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