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
To determine the masses \(m_1\) and \(m_2\), we analyze the forces in static equilibrium with the help of the given pulley system.
In this system, we have:
- A mass of 100 kg attached to the left-most pulley.
- Two other masses \(m_1\) and \(m_2\) attached to the system of pulleys and rope.
The tension in the rope throughout is consistent due to the frictionless nature of the pulleys. Let's define the tension in the rope as \(T\).
- For the 100 kg mass:
- The downward force due to gravity is \(100g\).
- Since the system is in equilibrium, the tension \(T\) in the rope must balance this force: \(T = 100g\).
- Analyzing the mass \(m_1\):
- Considering the tension in the rope, we have \(T = m_1 \times g\), so \(m_1 \times g = 100g\).
- Cancel out \(g\) from both sides to get \(m_1 = 100\) kg.
- Analyzing the mass \(m_2\):
- The tension in the rope supporting \(m_2\) and \(m_1\) is the same, so the effective tension here is shared between two strands of rope: \(T = 2T_1\).\)
- The tension in the rope at \(m_2\) is then \(T_1 = 50g\).\)
- This implies \(m_2 \times g = 50g\), and therefore, \(m_2 = 50\) kg.
The calculated values for the masses are \(m_1 = 100\) kg and \(m_2 = 50\) kg.
Conclusion: The correct option is 50, 100. Thus, the mass \(m_1 = 100\) kg and the mass \(m_2 = 50\) 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?
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.The approximate value of the integral
$\int_{2}^{3} \frac{dx}{x}$
using Simpson's rule with $h = 0.5$ is
