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Smugglers. Rumor has it that a company has been smuggling gold out of the country using sealed, cylindrical barrels with hollow walls. They pour molten gold into the hollows, and then fill the remainder of the barrel's internal volume with packing peanuts. The total mass of the gold-walled barrel was designed so that it exactly matches those used to transport a volatile chemical that cannot be exposed to air (and therefore the barrel cannot be opened and checked). The X-ray machine usually used to screen containers is suspiciously damaged and not available. (a) There are 20 barrels total, and they are all identical: mass \(m=50.0 \mathrm{kg}\), height \(h=1.20 \mathrm{m},\) and diameter \(D=0.25 \mathrm{m} .\) How do you determine which ones have walls filled with gold (and are essentially hollow on the interior except for packing peanuts) and those completely filled with the volatile chemical (a tightly-packed powder) where the mass is uniformly distributed? Hint: apply the concepts of moment of inertia. Assume that, in the case of the gold-filled barrels, the entire mass is concentrated at the outer wall of the barrel and, in the case of the barrels filled with the chemical, the mass is distributed evenly throughout the volume of the cylinder. You can neglect the circular bottoms and the lids of the barrels, and assume there is no slipping. (b) What is the acceleration of the center of mass of each of the barrels as they roll down a \(30^{\circ}\) inclined plane? (c) How much time does it take each barrel to roll \(10.0 \mathrm{m}\) down the \(30^{\circ}\) plane?

Step by step solution

01

Calculate the Moment of Inertia for Gold-filled and Chemical-filled Barrels

For the gold-filled barrels, the mass is concentrated at the outer edge, so use the moment of inertia for a hollow cylinder: \( I_{gold} = mR^2 \), where \( m = 50.0 \text{ kg} \) and \( R = \frac{D}{2} = 0.125 \text{ m} \).For the chemical-filled barrels, the mass is uniformly distributed, so use the moment of inertia for a solid cylinder: \( I_{chemical} = \frac{1}{2}mR^2 \).

02

Calculate the Acceleration on an Inclined Plane

The acceleration \( a \) of a rolling object down an incline without slipping can be calculated using the equation: \( a = \frac{g \sin \theta}{1 + \frac{I}{mR^2}} \), where \( I \) is the moment of inertia, \( g = 9.8 \text{ m/s}^2 \), and \( \theta = 30^{\circ} \).First, for a gold-filled barrel, substitute \( I_{gold} \) into the equation. Then, for a chemical-filled barrel, substitute \( I_{chemical} \) into the equation to calculate their respective accelerations.

03

Determine the Time to Roll Down the Incline

Use the kinematic equation \( s = \frac{1}{2}at^2 \) to find the time \( t \) it takes to travel \( s = 10.0 \text{ m} \) down the incline. Rearrange it to solve for \( t: t = \sqrt{\frac{2s}{a}} \).Calculate \( t \) using the accelerations found in Step 2 for both the gold-filled and chemical-filled barrels.

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Key Concepts

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

Solid Cylinder

A solid cylinder is a common shape in physics problems when dealing with rotational motion. In the case of a solid cylinder, the mass is evenly distributed throughout the volume.

For a solid cylinder such as the chemical-filled barrel in this exercise, the moment of inertia is calculated with the formula: \[ I = \frac{1}{2}mR^2 \]where:

• \( I \) is the moment of inertia,
• \( m \) is the mass of the cylinder, and
• \( R \) is the radius.

This concept is crucial to understand how the cylinder will roll down the incline because the moment of inertia directly affects the cylinder's acceleration. A solid cylinder will roll differently than a hollow one due to this distribution of mass, impacting the barrel's speed down the incline.

Hollow Cylinder

Hollow cylinders are distinct in that much of their material is located far from the central axis. In this problem, barrels with gold within the hollow walls resemble hollow cylinders.

The moment of inertia for a hollow cylinder, where mass is concentrated at the rim, is given by:\[ I = mR^2 \]Here, the entire mass contributes to the rotational inertia because it is all at the maximum radius from the axis of rotation.

• This means hollow cylinders have higher moments of inertia compared to solid cylinders of the same mass and radius, thus they resist changes to their rotational motion more.

This difference in distribution causes hollow cylinders to accelerate slower on an inclined plane, which can be a critical factor in identifying whether a barrel is filled with gold.

Rolling Motion

Rolling motion occurs when an object rolls without slipping on a surface. For cylinders on inclined planes, the combination of rotational and translational motion is essential to analyze.

In this exercise, the barrels roll down due to gravity, without slipping. Here, the relationship between translational acceleration \( a \) and moment of inertia \( I \) is given by:\[ a = \frac{g \sin \theta}{1 + \frac{I}{mR^2}} \]where:

• \( g \) is the acceleration due to gravity,
• \( \theta \) is the angle of the incline,
• \( I \) is the moment of inertia,
• \( m \) is the mass, and
• \( R \) is the radius.

The rolling motion illustrates how cylinders with different inertia values will affect their behavior when rolling. Solid and hollow cylinders will demonstrate this concept differently due to their distinct moments of inertia.

Inclined Plane

An inclined plane is a flat surface tilted at some angle to the horizontal. It's a classical physics problem setup to analyze motions under gravity.

In this scenario, the incline is set at \( 30^{\circ} \). This angle affects how gravity propels the rolling. The gravitational force can be split into components parallel and perpendicular to the incline. The force causing the roll is:\[ F = mg \sin \theta \]where:

• \( F \) is the force due to gravity,
• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity, and
• \( \theta \) is the incline angle.

The inclined plane causes different accelerations for barrels, depending on their moment of inertia. Thus, by observing how barrels roll down the incline, one can infer information about their internal structure, such as whether they contain gold or chemical.