Question

Two solid objects are made of different materials. Their volumes and volume expansion coefficients are \(V_{1}\) and \(V_{2}\) and \(\beta_{1}\) and \(\beta_{2}\), respectively. It is observed that during a temperature change of \(\Delta T\), the volume of each object changes by the same amount. If \(V_{1}=2 V_{2}\), what is the ratio of the volume expansion coefficients?

Short answer

Answer: The ratio of the volume expansion coefficients is $$\frac{\beta_{1}}{\beta_{2}} = \frac{1}{2}$$.

Step-by-step solution

Write the formula for volume expansion

The formula for volume expansion is given by: $$\Delta V = \beta \times V \times \Delta T$$ Where \(\Delta V\) is the change in volume, \(\beta\) is the volume expansion coefficient, \(V\) is the initial volume, and \(\Delta T\) is the temperature change.

Write the equation for each object

Since the volume of each object changes by the same amount, we can write the equation for each object and set them equal: $$\Delta V_{1} = \beta_{1} \times V_{1} \times \Delta T = \Delta V_{2} = \beta_{2} \times V_{2} \times \Delta T$$

Use the given information to simplify the equation

We are given that \(V_{1} = 2 V_{2}\). We can substitute this into our equation: $$\beta_{1} \times 2V_{2} \times \Delta T = \beta_{2} \times V_{2} \times \Delta T$$

Solve for the ratio

We need to find the value of $$\frac{\beta_{1}}{\beta_{2}}$$. To do this, first divide both sides of the equation by \(\Delta T\). $$\beta_{1} \times 2V_{2} = \beta_{2} \times V_{2}$$ Now, divide both sides of the equation by \(V_{2}\) and then divide both sides by \(\beta_{2}\): $$\frac{\beta_{1}}{\beta_{2}} = \frac{1}{2}$$ So, the ratio of the volume expansion coefficients is $$\frac{\beta_{1}}{\beta_{2}} = \frac{1}{2}$$.

Key concepts

Thermal Expansion

Think of thermal expansion as the reaction of a material when it is heated up: it expands! This is because the increase in temperature causes the atoms in the material to vibrate more and take up more space, leading to an increase in volume. For many materials, this expansion can be predicted and is often proportional to the original volume and the change in temperature.

Let's relate this to our exercise. We have two objects, each made of a different material. These different materials will likely have different responses to heat, which is described by the volume expansion coefficient, commonly denoted by \(\beta\). This coefficient tells us how much the volume of the material will increase per degree of temperature change. Simplifying it might sound like this: If you heat your chocolate bar, it will get a little bit larger; the degree to which it 'puffs up' with heat is its volume expansion coefficient.

Temperature Change in Physics

In physics, when we talk about temperature change denoted as \(\Delta T\), we're essentially discussing the difference between the initial and final temperature of a substance. This change is very important as it dictates a range of responses in materials, such as thermal expansion.

Heat something up, and its temperature increases; cool it down, and the temperature drops. Simple, right? But connecting this to our current task, the volume changes of our two objects are linked with the temperature change through a neat formula involving the volume expansion coefficients. This solidifies the concept that temperature change is one of the critical factors driving the physical expansion (or contraction) of materials.

Ratio of Physical Quantities

Ratios are everywhere, and in physics, they serve as a way to compare different physical quantities. For instance, when we say the ratio of \(\beta_{1}\) to \(\beta_{2}\) is \(\frac{1}{2}\), we mean that the volume expansion coefficient of the first material is half that of the second. Ratios give us a relative measure, helping us understand how one quantity relates to another.

In the exercise, by using simple algebra and the understanding of how ratios work, we can compare the volume expansion coefficients of our two objects. This comparison helps us predict how different materials behave under the same conditions without knowing their exact properties. Ratios like these are a fundamental tool in the physicist's toolkit, allowing for quick calculations and comparisons across various domains of physical science.