A bacterium can be approximated as a cylinder with a hemisphere at each end, as shown in the figure. The cylinder has a height of $1 \mu\text{m}$ and diameter of $1 \mu\text{m}$. Assuming that the density of the bacterium is equal to that of water, what is the approximate mass of this bacterium? Given: density of water = $10^3 \text{ kg/m}^3$; $1 \mu\text{m} = 10^{-6}\text{ m}$ Volume of a cylinder = $\pi r^2 h$, where $r$ is the radius and $h$ is the height of the cylinder Volume of a sphere = $\frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere
The bacterium can be approximated as a cylinder with a hemisphere at each end. We need to calculate the volume of this shape and subsequently find its mass, assuming the density is equal to that of water.
Step 1: Calculate the volume of the cylindrical part.
- The height of the cylinder, \(h = 1 \, \mu\text{m}\, = 10^{-6} \text{ m}\).
- The diameter = \(1 \, \mu\text{m}\, = 10^{-6} \text{ m}\), so the radius is \(r = 0.5 \times 10^{-6} \text{ m}\).
- Volume of the cylinder, \(V_{\text{cylinder}} = \pi r^2 h = \pi (0.5 \times 10^{-6})^2 \times 10^{-6} = 0.25 \pi \times 10^{-18} \text{ m}^3\).
Step 2: Calculate the volume of the hemispherical parts.
- Volume of a full sphere is \(V_{\text{sphere}} = \frac{4}{3} \pi r^3\).
- Volume of one hemisphere is half of this: \(V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3\).
- Total volume of both hemispheres: \(V_{\text{hemispheres}} = 2 \times \frac{2}{3} \pi (0.5 \times 10^{-6})^3 = \frac{4}{3} \pi \times 0.125 \times 10^{-18} \text{ m}^3 = \frac{\pi}{6} \times 10^{-18} \text{ m}^3\).
Step 3: Calculate the total volume of the bacterium.
- Total volume, \(V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemispheres}} = 0.25 \pi \times 10^{-18} + \frac{\pi}{6} \times 10^{-18} = \frac{\pi}{3} \times 10^{-18} \text{ m}^3\).
Step 4: Calculate the mass of the bacterium.
- Use the formula \(\text{mass} = \text{density} \times \text{volume}\).
- \(\text{density} = 10^3 \text{ kg/m}^3\).
- Mass = \(10^3 \times \frac{\pi}{3} \times 10^{-18} \text{ kg} = \frac{\pi}{3} \times 10^{-15} \text{ kg}\).
- Approximating \(\pi \approx 3.14\), the mass is around \(10^{-15} \text{ kg}\).
Conclusion: The approximate mass of the bacterium is \(10^{-15} \text{ kg}\). Thus, the correct answer is: \(10^{-15} \text{ kg}\).
Important Questions from Numerical Computation
- An organization allows its employees to work independently on consultancy projects but charges an overhead on the consulting fee. The overhead is 20% of the consulting fee, if the fee is up to . 5,00,000. For higher fees, the overhead is . 1,00,000 plus 10% of the amount by which the fee exceeds . 5,00,000. The government charges a Goods and Services Tax of 18% on the total amount (the consulting fee plus the overhead). An employee of the organization charges this entire amount, i.e., the consulting fee, overhead, and tax, to the client. If the client cannot pay more than . 10,00,000, what is the maximum consulting fee that the employee can charge?
Three frictionless pulleys with rope attachment are in a static equilibrium as shown in the figure. The mass $m_1$ and $m_2$, in kg, respectively are
If the in-situ density of coal is 1320 kg/m$^3$ and the density of blasted coal is 952 kg/m$^3$, the swell factor is _____________ (rounded off to 3 decimal places)
A five-member truss system is shown in the figure. The maximum vertical force P in kN that can be applied so that loads on the member CD and BC do NOT exceed 50 kN and 30 kN, respectively is _____________(rounded off to 2 decimal places)
The Fourier transform and its inverse transform are respectively defined as $\tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x)e^{i\omega x}dx$ and $f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega)e^{-i\omega x}d\omega$. Consider two functions $f$ and $g$. Another function $f * g$ is defined as
$(f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y)g(x - y)dy$
Which of the following relation is/are true?
Note: Tilde ($\sim$) denotes the Fourier transform.
