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Chapter 7: Problem 15

Calc (a) Describe how you would determine the center of mass of a uniform sheet of metal. (b) Describe how you would determine the center of mass if a small hole had been cut through the sheet. SSM

Short Answer

Expert verified
The center of mass of a uniform sheet of metal is its geometrical center. However, when a hole is cut out of it, the center of mass shifts towards the more massive part of the sheet, this is done by considering the hole as a negative mass.

Step by step solution

01

Center of Mass of a Uniform Sheet

For a uniform sheet, the mass density is the same everywhere. Therefore, the center of mass is simply the geometric center of the sheet. This can be found by drawing diagonals from the corners of the sheet, the point where the diagonals intersect will be the center of the sheet, thus the center of mass.
02

Center of Mass of Sheet with Hole

When a small hole is cut in the sheet, the mass distribution changes. To find the new center of mass, imagine the hole part as negative mass. The original center of mass location can be found as in Step 1. Then, determine the center of mass of the hole using the same technique, as if it's a separate object. Now calculate the COM of the 'combined' object, considering the hole part as having negative mass and the rest of the sheet as having positive mass. The center of mass will shift towards the more massive side.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass Density
In the realm of physics, mass density is a crucial aspect. It defines how much mass is present in a given volume or area of an object. When dealing with a uniform sheet of metal, the mass density is constant across the entire sheet. This makes calculations straightforward. The mass is distributed equally, meaning that every part of the sheet has the same "weight" in terms of how it affects the center of mass. Having a constant mass density is why the center of mass for uniform objects matches perfectly with the geometric center.
  • Consistent mass density implies equal mass spread.
  • Influences how the center of mass is calculated.
Remember, whenever mass distribution is uniform, the calculations are simpler due to uniform mass density.
Geometric Center
The geometric center is a specific point that comes into play when determining the center of mass, especially in uniform objects. For a symmetrical object like a rectangular sheet, this point is where the diagonals intersect.
  • It's the perfect midpoint of uniform objects.
  • In uniform mass density scenarios, it coincides with the center of mass.
To find this point in practice, just take a ruler and connect opposite corners. The intersecting point is your geometric center, thus the center of mass for objects with constant mass distribution.
Mass Distribution
Mass distribution refers to how mass is spread over a particular object or space. A uniform mass distribution means that mass is evenly spread throughout the object, as seen in parts of the original exercise with the metal sheet. But once we introduce changes, such as cutting a hole, the mass distribution becomes uneven.
  • Important for understanding shifts in the center of mass.
  • Uneven distribution requires more complex calculations for center of mass.
When dealing with uneven mass distributions, like with the sheet having a hole, the center of mass moves. The object no longer behaves as if it were uniform. Finding the new center of mass involves imagining the mass removed (hole) and how it changes the balance of the rest of the sheet.
Negative Mass
Negative mass is a fascinating concept in physics used for computational trickery. It's not something we can observe in the physical world, but it's a helpful model. In calculations like the one with the metal sheet hole, we treat the removed mass as negative to account for its absence.
  • Used to simplify complex calculations with missing parts.
  • Helps reposition the calculated center of mass by counteracting positive mass.
By understanding the hole as having negative mass, we balance the missing section with the remainder of the sheet. This shifts the center of mass toward the area with more mass, which gives us an accurate new center of mass for the altered object. It's like balancing a seesaw where the negative mass metaphorically "lifts up" the absent mass.

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Most popular questions from this chapter

Biology During mating season, male bighorn sheep establish dominance with head-butting contests which can be heard up to a mile away. When two males butt heads, the "winner" is the one that knocks the other backward. In one contest a sheep of a mass \(95 \mathrm{~kg}\) and moving at \(10 \mathrm{~m} / \mathrm{s}\) runs directly into a sheep of mass \(80 \mathrm{~kg}\) moving at \(12 \mathrm{~m} / \mathrm{s}\). Which ram wins the head-butting contest? SSM

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Calc A single force acts in the \(x\) direction on an object that is initially stationary. The force is described by \(F(t)=9 t^{2}+8 t^{3}\), where the constants carry SI units. Determine the momentum in the \(x\) direction of the object after the force has acted for \(1 \mathrm{~s} . \mathrm{SSM}\)

Two ice skaters, Lilly and John, face each other while stationary and push against each other's hands. John's mass is twice that of Lilly. How do their speeds compare after the push off? A. Lilly's speed is one-fourth of John's speed. B. Lilly's speed is one-half of John's speed. C. Lilly's speed is the same as John's speed. D. Lilly's speed is twice John's speed. E. Lilly's speed is four times John's speed.
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