Problem 20 - An Introduction to the Mechanics of Solids in SI Units

Question

The top of a tin can is removed, and the empty can is inverted over a pair of billiard balls on a table as shown in the sketch. For certain combinations of sizes and weights the configuration shown is stable. For other combinations the can tips over when released. It is proposed to set up a demonstration for a temperance lecture by using in sequence a frozen-orange-juice can and a beer can with the same pair of billiard balls. Investigate whether tipping will occur for the following sizes and weights. \(\begin{array}{lcl} & \text { Orange juice } & \text { Beer } \\ \text { Diameter of ball } & 45 \mathrm{~mm} & 45 \mathrm{~mm} \\ \text { Weight of ball } & 2.0 \mathrm{~N} & 2.0 \mathrm{~N} \\ \text { Diameter of can } & 50 \mathrm{~mm} & 70 \mathrm{~mm} \\ \text { Weight of empty can with lid removed } & 0.57 \mathrm{~N} & 1.0 \mathrm{~N}\end{array}\)

Step-by-Step Solution

Answer

Both the orange juice and the beer can will tip over when placed over the billiard balls.

Step 1: Calculate The Distance From The CoG To The Edge Of The Can

First, find the distance from the CoG of the system to the edge of the can. Since we only have the diameters, we must convert these to radius values by halving. So for the orange juice can it is \( \frac{50 mm}{2} = 25 mm \), and for the beer can \( \frac{70 mm}{2} = 35 mm \). The center of gravity(H) of the system will be the summation of the weight of the can times the height of the can (which is half its radius), and multiplied by 2 (because we have two balls with the same weight and radius), divided by the total weight of the system.

Step 2: Calculate The CoG For Orange Juice Can

Calculating H for the orange juice can, we have \( H = \frac{(0.57 N * 0.025 m) + (2 * 2.0 N * 0.045 m)}{0.57 N + 2 *2.0 N} = 0.0365 m = 36.5 mm \). The weight of the ball is greater than the weight of the juice can, hence, the centre of mass of the system is nearer to the centroid of the ball.

Step 3: Verify If The Orange Juice Can Will Tip Over

The orange juice can will tip over if the calculated H is greater than 25 mm (which is the radius of the base of the can). In this case, 36.5mm is greater than 25mm, so the orange juice can will tip over.

Step 4: Calculate The CoG For Beer Can

Calculating H for the beer can, we have \( H = \frac{(1.0 N * 0.035 m) + (2 * 2.0 N * 0.045 m)}{1.0 N + 2 *2.0 N} = 0.04125 m = 41.25 mm \). The weight of the ball is greater than the weight of the beer can, hence, the centre of mass of the system is nearer to the centre of the ball.

Step 5: Verify If The Beer Can Will Tip Over

The beer can will tip over if the calculated H is greater than 35 mm (which is the radius of the base of the can). In this case, 41.25 mm is greater than 35mm, so the beer can will tip over.

Key Concepts

Center of Gravity

In physics, the **center of gravity (CoG)** is a crucial concept that relates to the stability of objects. It is the point at which the entire weight of an object can be considered to act. When looking at an object like a tin can balanced on billiard balls, understanding where the CoG is located helps predict its stability.
The CoG can change depending on an object's weight distribution. If the CoG is aligned over the base of support, such as a pair of billiard balls, the object should remain stable. However, if the CoG shifts beyond this base, the object will tip over. In our example, you need to calculate the CoG of a system to determine how it behaves when balanced on the balls. This calculation involves finding the distance of the CoG from the edge of the can, which helps us understand the object's balance and stability.

Tipping Analysis

**Tipping analysis** involves evaluating whether an object will remain upright or tip over. In our exercise, tipping occurs if the center of gravity (CoG) goes beyond the edge of a can when the cans are inverted over the billiard balls.
To perform tipping analysis:

• Determine the radius of the can's base, as any displacement of the CoG beyond this point signifies instability.
• Calculate the position of the CoG. If the CoG lies outside the radius of the can’s base, tipping will occur.

For the orange juice can, 36.5 mm is the height of the CoG, which is greater than the radius (25 mm), leading to tipping. Similarly, for the beer can, with a CoG height of 41.25 mm, which exceeds its radius (35 mm), tipping is inevitable. This analysis ensures that any imbalance due to disproportionate weight distribution is accurately assessed, preventing unwanted movements.

Weight Distribution

**Weight distribution** describes how weight is spread out over an object. It directly influences the center of gravity (CoG) and, ultimately, the stability of the system in question.
For stable setups, weight needs to be well-distributed around the CoG. In our demonstration:

• The billiard balls, each weighing 2.0 N, are a significant part of the system, tipping the balance greatly due to their weight relative to the cans.
• The relatively lighter empty cans (0.57 N for the orange juice can and 1.0 N for the beer can) have their centers of gravity pulled closer to the heavier billiard balls’ positions.

Thus, in such situations, the CoG is closer to the balls, resulting in an increased risk of tipping unless measures are taken to rebalance the weight or adjust the base width. Correctly understanding weight distribution can help engineers and designers predict and counteract potential stability issues in real-world applications.

Physics Demonstration

A **physics demonstration** serves as an educational tool to visually explain theoretical concepts. In this exercise, the demonstration showcasing the stability of cans on billiard balls can be used to understand tipping points and balance.
The goal is to exhibit principles like:

• How the center of gravity affects stability.
• The impact of weight distribution on balance.
• The point at which an object tips over when supported inadequately.

This simple yet effective demonstration helps participants visualize abstract ideas in tangible forms, fortifying their grasp on principles of mechanics. By progressively analyzing different weight and size configurations, observers can see firsthand how subtle changes influence the stability of systems. Additionally, such experiments demystify complex concepts and encourage learners to think critically about the dynamics of equilibrium and motion.