Problem 42 - College Physics

Question

An object weighing \(300 \mathrm{~N}\) in air is immersed in water after being tied to a string connected to a balance. The scale now reads \(265 \mathrm{~N}\). Immersed in oil, the object appears to weigh \(275 \mathrm{~N}\). Find (a) the density of the object and (b) the density of the oil.

Step-by-Step Solution

Answer

The density of the object is approximately \(8577.038 kg/m^3\) and the density of the oil is approximately \(699.159 kg/m^3\).

Step 1: Find Effective Weights

First, determine the effective weights of the object in different fluids by subtracting the weight of the object in the fluid from the weight of the object in air. The effective weight in water is \(265 N - 300 N = - 35 N\) and in oil is \(275 N - 300 N = - 25 N\). The negative sign indicates that the buoyant force is acting upwards.

Step 2: Finding the Volume

The volume of the object can be found using the effective weight in water and the density of water, \(1000 kg/m^3\). Use the formula of buoyant force, \(F_b = \rho_{fluid} \cdot V \cdot g\), where \(F_b = -35 N\), \(g = 9.8 m/s^2\), and solve for \(V\): \(V = - F_b / (\rho_{fluid}\cdot g) = -(-35N) / (1000 kg/m^3 \cdot 9.8 m/s^2) = 0.00357 m^3 \).

Step 3: Finding the Density of the Object

Next, find the density of the object, denoted as \(ρ_{object}\), using the formula \(ρ = m/v\), where \(m\) is the mass and \(v\) is the volume. As weight \(W = m \cdot g\), the mass \(m = W/g = 300N / 9.8 m/s^2 = 30.612 kg\). Substituting the values into the density formula gives: \(ρ_{object} = m / v = 30.612 kg / 0.00357 m^3 = 8577.038 kg/m^3\).

Step 4: Finding the Density of the Oil

Considering the buoyancy in oil, the density of the oil can be calculated by rearranging the formula of buoyant force to get \(ρ_{fluid} = - F_b / (V \cdot g)\). Substituting the known values gives: \(\rho_{oil} = -(-25N) / (0.00357 m^3 \cdot 9.8 m/s^2) = 699.159 kg/m^3\).