A sinker of weight w0 has an apparent weight w1 when placed in a liquid at a temperature T1 and w2 w

Q: Why does the apparent weight of the sinker change with temperature?

As temperature increases, both the liquid and the sinker expand. The liquid's density decreases, which reduces the upthrust. However, the sinker's volume increases, which tends to increase the upthrust. The net change in apparent weight depends on which expansion effect dominates.

Q: Why do we use the formula $\rho_T = \rho_0 / (1 + \gamma \Delta T)$ instead of $\rho_T = \rho_0 (1 - \gamma \Delta T)$?

The formula $\rho_T = \rho_0 / (1 + \gamma \Delta T)$ is exact because density is inversely proportional to volume ($V_T = V_0(1 + \gamma \Delta T)$). The linear approximation $\rho_0 (1 - \gamma \Delta T)$ is only valid for very small temperature changes, and using the exact form prevents algebraic errors in derivations.

Q: Can the apparent weight at a higher temperature be less than at a lower temperature?

Yes! If the sinker expands much more than the liquid (which is rare but theoretically possible), the displaced volume increases faster than the liquid's density drops, leading to a greater upthrust and a lower apparent weight.

Q: What if the sinker's material doesn't expand at all ($\beta = 0$)?

If the sinker doesn't expand, its volume remains constant. The change in upthrust would then be entirely due to the decrease in the liquid's density. The formula simplifies beautifully to $\beta_l = \frac{w_2 - w_1}{(w_0 - w_2)(T_2 - T_1)}$.

Q: Why did we divide the two density ratio equations instead of subtracting them?

Dividing the equations elegantly eliminates the true weight $w_0$ from the denominators on the right-hand side, creating a clean ratio of densities that directly maps to the temperature expansion formulas. Subtracting them would lead to a messy algebraic swamp!

Question

A sinker of weight $w_0$ has an apparent weight $w_1$ when placed in a liquid at a temperature $T_1$ and $w_2$ when weighed in the same liquid at a temperature $T_2$. The coefficient of cubical expansion of the material of the sinker is $\beta$. What is the coefficient of volume expansion of the liquid ?

Sigma Percentile · Accepted Answer

### The Beauty of Archimedes Meets Thermodynamics

Imagine you are in a laboratory, tasked with finding out exactly how much a mysterious liquid expands when heated. You don't have fancy electronic sensors, just a solid metal sinker, a spring balance, and a thermometer. How do you do it?

This classic problem beautifully marries the mechanics of Archimedes' principle with the thermodynamics of thermal expansion. By simply measuring the apparent weight of the sinker at two different temperatures, we can unlock the thermal secrets of the liquid. Let's dive into the elegant math behind this experiment.

### Analyzing the Setup: Apparent Weight and Upthrust

When a sinker of true weight $w_0$ is completely immersed in a liquid, it feels lighter. This is due to the upward buoyant force, or upthrust ($F_B$), exerted by the liquid. The spring balance reads this reduced weight, which we call the apparent weight ($w_{app}$).

$$w_{app} = w_0 - F_B$$

According to Archimedes, this upthrust equals the weight of the displaced liquid. Mathematically, $F_B = V ho_l g$, where $V$ is the volume of the sinker and $ho_l$ is the density of the liquid.

Since the volume of the sinker can be written as its mass divided by its density ($V = \frac{m}{ho_s} = \frac{w_0}{ho_s g}$), we can express the upthrust entirely in terms of weights and densities:

$$F_B = \left( \frac{w_0}{ho_s g} ight) ho_l g = w_0 \frac{ho_l}{ho_s}$$

### The Master Equations at Two Temperatures

Now, let's apply this logic to our two temperature states. At the initial temperature $T_1$, the apparent weight is $w_1$. The densities of both the liquid and the sinker are specific to this temperature.

$$w_1 = w_0 - w_0 \left[ \frac{(ho_l)_{T_1}}{(ho_s)_{T_1}} ight]$$

Similarly, when we heat the system to a higher temperature $T_2$, both the liquid and the sinker expand. Their densities drop, altering the upthrust and giving us a new apparent weight, $w_2$.

$$w_2 = w_0 - w_0 \left[ \frac{(ho_l)_{T_2}}{(ho_s)_{T_2}} ight]$$

### The Algebraic Dance: Isolating Density Ratios

Our goal is to find the expansion coefficient of the liquid, which is hidden inside those density terms. Let's rearrange our two equations to isolate the ratio of liquid density to sinker density at each temperature.

$$\frac{(ho_l)_{T_1}}{(ho_s)_{T_1}} = \frac{w_0 - w_1}{w_0}$$

$$\frac{(ho_l)_{T_2}}{(ho_s)_{T_2}} = \frac{w_0 - w_2}{w_0}$$

To connect these two states and eliminate the pesky $w_0$ from the denominators, we divide the first equation by the second. This is a strategic algebraic move that creates a clean ratio of densities.

$$\frac{(ho_l)_{T_1}}{(ho_l)_{T_2}} 	imes \frac{(ho_s)_{T_2}}{(ho_s)_{T_1}} = \frac{w_0 - w_1}{w_0 - w_2}$$

### Introducing Thermal Expansion

Now we bring in thermodynamics. We know that as a substance heats up, its volume increases, causing its density to decrease according to the relation $ho_T = \frac{ho_0}{1 + \gamma \Delta T}$.

Using this, the ratio of the liquid's density at $T_1$ to its density at $T_2$ is simply $1 + \beta_l (T_2 - T_1)$, where $\beta_l$ is the volume expansion coefficient of the liquid.

Similarly, for the sinker, the ratio of its density at $T_2$ to its density at $T_1$ is $\frac{1}{1 + \beta_s (T_2 - T_1)}$. The problem states that the sinker's expansion coefficient $\beta_s$ is simply $\beta$.

Substituting these expansion terms into our master ratio equation, we get:

$$\frac{1 + \beta_l (T_2 - T_1)}{1 + \beta (T_2 - T_1)} = \frac{w_0 - w_1}{w_0 - w_2}$$

### Final Calculation: Isolating the Liquid's Secret

We are in the endgame now. We just need to isolate $\beta_l$. Let's cross-multiply to bring the term containing $\beta_l$ to one side.

$$1 + \beta_l (T_2 - T_1) = \left( \frac{w_0 - w_1}{w_0 - w_2} ight) [1 + \beta (T_2 - T_1)]$$

Expanding the right side and moving the $1$ over:

$$\beta_l (T_2 - T_1) = \left( \frac{w_0 - w_1}{w_0 - w_2} ight) - 1 + \beta \left( \frac{w_0 - w_1}{w_0 - w_2} ight) (T_2 - T_1)$$

Notice that $\frac{w_0 - w_1}{w_0 - w_2} - 1$ simplifies beautifully to $\frac{w_2 - w_1}{w_0 - w_2}$. Finally, dividing the entire equation by the temperature difference $(T_2 - T_1)$, we arrive at our grand result:

**$$\beta_l = \beta \left( \frac{w_0 - w_1}{w_0 - w_2} ight) + \frac{w_2 - w_1}{(w_0 - w_2)(T_2 - T_1)}$$**

This elegant formula shows exactly how the liquid's expansion is a delicate balance of the sinker's own expansion and the shift in the measured apparent weights. It is a testament to how fundamental physics principles can be woven together to measure the invisible properties of matter!

Visualized Solution

Visualizing the Setup

Apparent Weight and Upthrust

Expressing Upthrust

State at Temperature $T_{1}$

State at Temperature $T_{2}$

Isolating Density Ratios

Relating the Two States

Density Variation with Temperature

Substituting Expansion Terms

Rearranging for $β_{l}$

Final Calculation

The Way Forward

The Sigma Insight: Thermal Expansion

The Beauty of Archimedes Meets Thermodynamics

Analyzing the Setup

The Master Equations at Two Temperatures

The Algebraic Dance

Introducing Thermal Expansion

Final Calculation

Similar Questions

Two different wires having lengths $L_{1}$ and $L_{2}$ and respective temperature coefficients of linear expansion $α_{1}$ and $α_{2}$ , are joined end-to-end. Then the effective temperature coefficient of linear expansion is

Visualized Solution

Visualizing the Setup

Apparent Weight and Upthrust

Expressing Upthrust

State at Temperature .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T1​

State at Temperature .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T2​

Isolating Density Ratios

Relating the Two States

Density Variation with Temperature

Substituting Expansion Terms

Rearranging for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } βl​

Final Calculation

The Way Forward

The Sigma Insight: Thermal Expansion

The Beauty of Archimedes Meets Thermodynamics

Analyzing the Setup

The Master Equations at Two Temperatures

The Algebraic Dance

Introducing Thermal Expansion

Final Calculation

Similar Questions

State at Temperature $T_{1}$

State at Temperature $T_{2}$

Rearranging for $β_{l}$