Problem 20 - Master Resource Book in JEE Main Physics

Question

An alloy of \(\mathrm{Zn}\) and Cu (i.e., brass) weights \(16.8 \mathrm{~g}\) in air and \(14.7 \mathrm{~g}\) in water. If relative density of \(\mathrm{Cu}\) and \(\mathrm{Zn}\) are \(8.9\) and \(7.1\) respectively then determine the amount of \(\mathrm{Zn}\) and \(\mathrm{Cu}\) in the alloy. (a) \(2 \mathrm{~g} .4 \mathrm{~g}\) (b) \(4 \mathrm{~g}, 2 \mathrm{~g}\) (c) 9.345g. \(7.455 \mathrm{~g}\) (d) \(0,3 \mathrm{~g}\)

Step-by-Step Solution

Answer

The masses of Zn and Cu are 7.455 g and 9.345 g, respectively.

Step 1: Calculate Buoyant Force

First, calculate the buoyant force experienced by the alloy when submerged in water. The buoyant force is equal to the weight of the water displaced, which can be found by subtracting the weight of the alloy in water from its weight in air: \[ F_{b} = 16.8 \, \mathrm{g} - 14.7 \, \mathrm{g} = 2.1 \, \mathrm{g} \]

Step 2: Find Volume of Alloy

Since the density of water is \(1 \mathrm{~g/cm^3}\), the volume of the alloy is equal to the buoyant force in grams. Thus, \[ V_{alloy} = 2.1 \, \mathrm{cm^3} \]

Step 3: Calculate Density of Alloy

Next, calculate the overall density of the alloy by dividing its mass in air by its volume:\[ \text{Density of Alloy} = \frac{16.8 \, \mathrm{g}}{2.1 \, \mathrm{cm^3}} = 8.0 \, \mathrm{g/cm^3} \]

Step 4: Set Up System of Equations

Let \( x \) be the mass of Zinc (\( \mathrm{Zn} \)) and \( y \) be the mass of Copper (\( \mathrm{Cu} \)). We have the following equations from the problem description:1. \( x + y = 16.8 \) 2. \( \frac{x}{7.1} + \frac{y}{8.9} = 2.1 \) Equation 1 represents the total mass, and Equation 2 represents the total volume.

Step 5: Solve System of Equations

Let's solve the system of linear equations starting with substituting Equation 1 into Equation 2. Substitute \( y = 16.8 - x \) into Equation 2:\[\frac{x}{7.1} + \frac{16.8 - x}{8.9} = 2.1\]Solve for \( x \):First, clear the fractions by multiplying through by the common denominator, which is \(7.1 \times 8.9\):\[8.9x + 7.1(16.8 - x) = 2.1 \times 7.1 \times 8.9\]Simplify:\[8.9x + 119.28 - 7.1x = 132.711\]Combine \( x \) terms:\[1.8x = 13.431\]Solve for \( x \):\[x \approx 7.455 \mathrm{~g}\]Now, substitute \( x = 7.455 \) back into Equation 1 to find \( y \):\[ y = 16.8 - 7.455 = 9.345 \mathrm{~g}\]

Step 6: Check Solution

Recalculate the volumes of \( \mathrm{Zn} \) and \( \mathrm{Cu} \) using their densities to verify:\[\frac{7.455}{7.1} + \frac{9.345}{8.9} \approx 2.1\]This confirms the solution is correct. Thus, the masses of \( \mathrm{Zn} \) and \( \mathrm{Cu} \) in the alloy are \( 7.455 \mathrm{~g} \) and \( 9.345 \mathrm{~g} \), respectively.

Key Concepts

Buoyant Force

When an object is submerged in a fluid, it experiences an upward force known as the buoyant force. This force arises because the fluid exerts pressure on the object from all directions, and the pressure at the bottom of the object is greater than the pressure at the top. Thus, there's a net upward force. For our exercise, the alloy of zinc (Zn) and copper (Cu) experienced a buoyant force while submerged in water. We calculated this force by subtracting the weight of the alloy in water (14.7 g) from its weight in air (16.8 g), giving us a buoyant force of 2.1 g. This number indicates the weight of the water displaced by the alloy, and underpins our later calculations, such as determining the alloy's volume.

Density Calculation

Density is a measure of mass per unit volume and is a crucial factor in understanding material composition. In this exercise, we were dealing with an alloy with a total weight of 16.8 g in air. Knowing the volume was key, and we used the buoyant force to determine this since the volume of the displaced water equals the volume of the alloy. Given the density of water is 1 g/cm³, we found the volume to be 2.1 cm³, which is numerically equal to the buoyant force we calculated earlier. By dividing the mass in air (16.8 g) by the volume (2.1 cm³), we determined the density of the alloy to be 8.0 g/cm³. This density information allowed us to set up further calculations to discern the proportions of zinc and copper in the alloy.

System of Linear Equations

Solving systems of linear equations is a critical method in mathematics for finding unknown variables. In this exercise, we used it to find the individual masses of zinc and copper in the alloy. We established two equations based on the problem's conditions:

• The sum of the masses of zinc and copper (Equation 1): \( x + y = 16.8 \)
• The sum of their respective volumes derived from their densities (Equation 2): \( \frac{x}{7.1} + \frac{y}{8.9} = 2.1 \)

We substituted Equation 1 into Equation 2 to isolate one variable, which enabled us to solve the system. By substitution, we found the mass of zinc (\( x \)) to be approximately 7.455 g and the mass of copper (\( y \)) to be 9.345 g. Solving these equations verified the accuracy of our solution and confirmed the individual metal proportions in the alloy.