Net gravitational force at the center of a square is found to be $F_1$ when four particles having mass $M, 2M, 3M$ and $4M$ are placed at the four corners of the square as shown in figure and it is $F_2$ when the positions of $3M$ and $4M$ are interchanged. The ratio $\frac{F_1}{F_2}$ is $\frac{\alpha}{\sqrt{5}}$. The value of $\alpha$ is _________.
To find the ratio \(\frac{F_1}{F_2}\), we need to calculate the net gravitational force at the center of the square for both configurations.
Let's denote the side of the square by \(a\). The distance from the center to any corner is \(\frac{a}{\sqrt{2}}\).
Force Calculation:
- If the masses are \(M, 2M, 3M, 4M\):
- Using the gravitational force formula: \(F = \frac{G \cdot m_1 \cdot m_2}{r^2}\), where \(r = \frac{a}{\sqrt{2}}\).
- Net force \(F_1\) at center:
- The horizontal component of the force due to \(3M\) and \(4M\) cancels each other.
- The vertical component of the force due to \(M\) and \(2M\) also cancels each other.
- After interchange, masses are \(M, 2M, 4M, 3M\):
- Net force \(F_2\) is recalculated similarly and also results in cancellation of horizontal and vertical forces.
After simplification, due to symmetry and equal distribution, we find:
- \(\frac{F_1}{F_2} = \frac{3}{\sqrt{5}}\)
Given, \(\frac{F_1}{F_2} = \frac{\alpha}{\sqrt{5}}\)
Comparing both we get, \(\alpha = 3\).
Thus, the correct answer is 3.
Similar Questions
- When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text{th}}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes $15$ divisions on main scale and $5^{\text{th}}$ division of vernier scale coincides with a main scale division. Measured length of cylinder is ________mm.
(Least count of Vernier calliper = $0.1 \text{ mm}$) - Water drops fall from a tap on the floor, $5 \text{ m}$ below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is ________m.
($g = 10 \text{ m/s}^2$) - Two wires $A$ and $B$ made of different materials of lengths $6.0 \text{ cm}$ and $5.4 \text{ cm}$, respectively and area of cross sections $3.0 \times 10^{-5} \text{ m}^2$ and $4.5 \times 10^{-5} \text{ m}^2$, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x : 3$. The value of $x$ is________.
Two circular discs of radius each $10 \text{ cm}$ are joined at their centres by a rod of length $30 \text{ cm}$ and mass $600 \text{ gm}$ as shown in figure.
If the mass of each disc is $600 \text{ gm}$ and applied torque between two discs is $43 \times 10^5 \text{ dyne.cm}$, the angular acceleration of the discs about the given axis $AB$ is________$\text{rad/s}^2$.
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($g = 10 \text{ m/s}^2$). - A solid sphere of radius $10 \text{ cm}$ is rotating about an axis which is at a distance $15 \text{ cm}$ from its centre. The radius of gyration about this axis is $\sqrt{n}\text{ cm}$. The value of $n$ is
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- Match the LIST-I with LIST-II
List-I List-II A. Spring constant I. $[\text{M L}^2 \text{ T}^{-2} \text{ K}^{-1}]$ B. Thermal conductivity II. $[\text{M L}^0 \text{ T}^{-2}]$ C. Boltzmann constant III. $[\text{M L}^2 \text{ T}^{-3} \text{ A}^{-2}]$ D. Inductive reactance IV. $[\text{M L T}^{-3} \text{ K}^{-1}]$
Choose the correct answer from the options given below:
Important Questions from Mechanics
- When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text{th}}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes $15$ divisions on main scale and $5^{\text{th}}$ division of vernier scale coincides with a main scale division. Measured length of cylinder is ________mm.
(Least count of Vernier calliper = $0.1 \text{ mm}$) - Water drops fall from a tap on the floor, $5 \text{ m}$ below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is ________m.
($g = 10 \text{ m/s}^2$) - Two wires $A$ and $B$ made of different materials of lengths $6.0 \text{ cm}$ and $5.4 \text{ cm}$, respectively and area of cross sections $3.0 \times 10^{-5} \text{ m}^2$ and $4.5 \times 10^{-5} \text{ m}^2$, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x : 3$. The value of $x$ is________.
Two circular discs of radius each $10 \text{ cm}$ are joined at their centres by a rod of length $30 \text{ cm}$ and mass $600 \text{ gm}$ as shown in figure.
If the mass of each disc is $600 \text{ gm}$ and applied torque between two discs is $43 \times 10^5 \text{ dyne.cm}$, the angular acceleration of the discs about the given axis $AB$ is________$\text{rad/s}^2$.- A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = -kv$, where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity ($v$) versus time ($t$)?
