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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

You find yourself in the (realistic?) situation of being stuck on a 300 -kg raft (including yourself) in the middle of a pond with nothing but a pile of 7 -kg bowling balls and 55 -g tennis balls. Using your knowledge of rocket propulsion, you decide to start throwing balls from the raft to move toward shore. Which of the following will allow you to reach the shore faster? a) throwing the tennis balls at \(35 \mathrm{~m} / \mathrm{s}\) at a rate of 1 tennis ball per second b) throwing the bowling balls at \(0.5 \mathrm{~m} / \mathrm{s}\) at a rate of 1 bowling ball every \(3 \mathrm{~s}\) c) throwing a tennis ball and a bowling ball simultaneously, with the tennis ball moving at \(15 \mathrm{~m} / \mathrm{s}\) and the bowling ball moving at \(0.3 \mathrm{~m} / \mathrm{s}\), at a rate of 1 tennis ball and 1 bowling ball every \(4 \mathrm{~s}\) d) not enough information to decide

Find the location of the center of mass for a onedimensional rod of length \(L\) and of linear density \(\lambda(x)=c x\), where \(c\) is a constant. (Hint: You will need to calculate the mass in terms of \(c\) and \(L\).)

A student with a mass of \(40.0 \mathrm{~kg}\) can throw a \(5.00-\mathrm{kg}\) ball with a relative speed of \(10.0 \mathrm{~m} / \mathrm{s}\). The student is standing at rest on a cart of mass \(10.0 \mathrm{~kg}\) that can move without friction. If the student throws the ball horizontally, what will the velocity of the ball with respect to the ground be?

A circular pizza of radius \(R\) has a circular piece of radius \(R / 4\) removed from one side, as shown in the figure. Where is the center of mass of the pizza with the hole in it?

A projectile is launched into the air. Part way through its flight, it explodes. How does the explosion affect the motion of the center of mass of the projectile?
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