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Chapter 8: Problem 8

Equal volumes of water and alcohol are mixed together. The density of water is \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) and the density of alcohol is \(800 \mathrm{~kg} / \mathrm{m}^{3}\). The density of the mixture is (A) \(900 \mathrm{~kg} / \mathrm{m}^{3}\) (B) \(1100 \mathrm{~kg} / \mathrm{m}^{3}\) (C) \(875 \mathrm{~kg} / \mathrm{m}^{3}\) (D) \(950 \mathrm{~kg} / \mathrm{m}^{3}\)

Short Answer

Expert verified
The density of the mixture is \(900 \mathrm{~kg} / \mathrm{m}^{3}\) (option A).

Step by step solution

01

Calculate Masses of Water and Alcohol

To find the mass of each component, we can use the formula: mass = density 脳 volume Since we have equal volumes of water and alcohol, let's consider their volumes to be V. Now, the mass of water (m_w) and alcohol (m_a) can be determined as follows: \(m_w = 1000 \mathrm{~kg} / \mathrm{m}^{3} \times V \) \(m_a = 800 \mathrm{~kg} / \mathrm{m}^{3} \times V \)
02

Calculate Total Mass of the Mixture

Add the masses of water and alcohol together to calculate the total mass (m_total) of the mixture: \( m_{total} = m_w + m_a \)
03

Calculate Total Volume of the Mixture

Since we have equal volumes of water and alcohol, the total volume (V_total) of the mixture is: \( V_{total} = V + V = 2V \)
04

Calculate the Density of the Mixture

To find the density of the mixture, divide the total mass by the total volume: \(Density = \frac{m_{total}}{V_{total}}\) Substitute the values of total mass and total volume: \(Density = \frac{(1000 \mathrm{~kg} / \mathrm{m}^{3} \times V) + (800 \mathrm{~kg} / \mathrm{m}^{3} \times V)}{2V}\) Simplify the expression by dividing the common factor 'V': \(Density = \frac{1000 + 800}{2}~\mathrm{kg}/\mathrm{m}^3\)
05

Calculate the Final Density

Finish the calculation to find the final density: \(Density = \frac{1800}{2}~\mathrm{kg}/\mathrm{m}^3 = 900 \mathrm{~kg} / \mathrm{m}^{3}\) The density of the mixture is \(900 \mathrm{~kg} / \mathrm{m}^{3}\), which corresponds to option (A).

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Most popular questions from this chapter

An air bubble of radius \(r\) in water is at a depth \(h\) below the water surface at some instant. If \(P\) is atmospheric pressure and \(d\) and \(T\) are the density and surface tension of water, respectively, the pressure inside the bubble will be (A) \(P+h d g-\frac{4 T}{r}\) (B) \(P+h d g+\frac{2 T}{r}\) (C) \(P+h d g-\frac{2 T}{r}\) (D) \(P+h d g+\frac{4 T}{r}\)

A wire suspended vertically from one of its ends is stretched by attaching a weight of \(200 \mathrm{~N}\) to the lower end. The weight stretches the wire by \(1 \mathrm{~mm}\). Then, the elastic energy stored in the wire is \([2003]\) (A) \(0.2 \mathrm{~J}\) (B) \(10 \mathrm{~J}\) (C) \(20 \mathrm{~J}\) (D) \(0.1 \mathrm{~J}\)

An open vessel containing water is given a constant acceleration \(a\) in the horizontal direction. Then the free surface of water gets sloped with the horizontal at an angle \(\theta\) given by (A) \(\theta=\tan ^{-1}\left(\frac{a}{g}\right)\) (B) \(\theta=\tan ^{-1}\left(\frac{g}{a}\right)\) (C) \(\theta=\sin ^{-1}\left(\frac{a}{g}\right)\) (D) \(\theta=\cos ^{-1}\left(\frac{g}{a}\right)\)

A tank, which is open at the top, contains a liquid up to a height \(H\). A small hole is made in the side of the tank at a distance \(y\) below the liquid surface. The liquid emerging from the hole lands at a distance \(x\) from the \(\operatorname{tank}\) (A) If \(y\) is increased from zero to \(H, x\) will first increase and then decrease. (B) \(x\) is maximum for \(y=H / 2\). (C) The maximum value of \(x\) is \(H\). (D) The maximum value of \(x\) will depend on the density of the liquid.

A vessel of \(1 \times 10^{-3} \mathrm{~m}^{3}\) volume contains oil, when a pressure of \(1.2 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\) is applied on it, then volume decreases by \(0.3 \times 10^{-6} \mathrm{~m}^{3}\). The bulk modulus of oil is (A) \(1 \times 10^{6} \mathrm{~N} / \mathrm{m}^{2}\) (B) \(2 \times 10^{7} \mathrm{~N} / \mathrm{m}^{2}\) (C) \(4 \times 10^{8} \mathrm{~N} / \mathrm{m}^{2}\) (D) \(6 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}\)
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