(Gravitational constant $G = 6.7 \times 10^{-11} \text{ N m}^2/\text{kg}^2$)
Initial Potential Energy Calculation
The work done is equal to the change in gravitational potential energy: $W = U_{final} - U_{initial}$.
The gravitational potential energy ($U$) for three masses ($m_1, m_2, m_3$) at the vertices of an equilateral triangle with side length $a$ is given by:
$ U = -G \left( \frac{m_1 m_2}{a} + \frac{m_1 m_3}{a} + \frac{m_2 m_3}{a} \right) = -\frac{G}{a} (m_1 m_2 + m_1 m_3 + m_2 m_3) $
Given masses are $m_1 = 200 \text{ kg}$, $m_2 = 300 \text{ kg}$, and $m_3 = 400 \text{ kg}$.
Calculate the product sum:
- $m_1 m_2 = 200 \times 300 = 60000 \text{ kg}^2$
- $m_1 m_3 = 200 \times 400 = 80000 \text{ kg}^2$
- $m_2 m_3 = 300 \times 400 = 120000 \text{ kg}^2$
- Sum $= 60000 + 80000 + 120000 = 260000 \text{ kg}^2$
For the initial state, the side length $a_1 = 20 \text{ m}$.
$ U_{initial} = -\frac{G}{20 \text{ m}} (260000 \text{ kg}^2) = -13000 G \text{ J} $
Final Potential Energy Calculation
For the final state, the side length $a_2 = 25 \text{ m}$.
$ U_{final} = -\frac{G}{25 \text{ m}} (260000 \text{ kg}^2) = -10400 G \text{ J} $
Work Done Determination
Calculate the work done using the change in potential energy:
$ W = U_{final} - U_{initial} $
$ W = (-10400 G) - (-13000 G) $
$ W = (-10400 + 13000) G $
$ W = 2600 G \text{ J} $
Substitute the value of the gravitational constant $G = 6.7 \times 10^{-11} \text{ N m}^2/\text{kg}^2$:
$ W = 2600 \times (6.7 \times 10^{-11}) \text{ J} $
$ W = 17420 \times 10^{-11} \text{ J} $
$ W = 1.742 \times 10^{-7} \text{ J} $
Rounding to two decimal places for comparison with options gives $1.74 \times 10^{-7} \text{ J}$.
Similar Questions
- When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
A uniform bar of length 12 cm and mass $20m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is :
- Two masses $m$ and $2m$ are connected by a light string going over a pulley (disc) of mass $30m$ with radius $r=0.1 \text{ m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The $2m$ mass is released from rest and its speed when it has descended through a height of 3.6 m is _________ m/s. (Assume string does not slip and $g = 10 \text{ m/s}^2$)
- A spring of force constant 15 N/m is cut into two pieces. If the ratio of their length is 1:3, then the force constant of smaller piece is ______ N/m.
Match the LIST-I with LIST-II
List-I: List-II: A. Magnetic induction I. $[M L T^{-2} A^{-2}]$ B. Magnetic flux II. $[M L^2 T^{-2} A^{-2}]$ C. Magnetic permeability III. $[M L^0 T^{-2} A^{-1}]$ D. Self inductance IV. $[M L^2 T^{-2} A^{-1}]$ Choose the correct answer from the options given below:
- Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm. When released from rest the heavier mass is observed to fall 81 cm in 9 s. The rotational inertia of the pulley is ______ $\text{kg.m}^2$. ($g = 9.8 \text{ m/s}^2$)
In the given figure the blocks $A$, $B$ and $C$ weigh 4 kg, 6 kg and 8 kg respectively. The co-efficient of sliding friction between any two surfaces is 0.5. The force $\vec{F}$ required to slide the block $C$ with constant speed is ______ N. (Use $g = 10 \text{ m/s}^2$)
In case of vertical circular motion of a particle by a thread of length $r$ if the tension in the thread is zero at an angle $30^\circ$ shown in figure, the velocity at the bottom point ($A$) of the circular path is
($g = \text{gravitational acceleration}$)
- The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is $5/x$. The value of $x$ is _______.
- In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division $= 0.05 \text{ mm}$, then the least count of the vernier callipers is _______ mm.
Important Questions from Mechanics
- When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
A uniform bar of length 12 cm and mass $20m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is :
- Two masses $m$ and $2m$ are connected by a light string going over a pulley (disc) of mass $30m$ with radius $r=0.1 \text{ m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The $2m$ mass is released from rest and its speed when it has descended through a height of 3.6 m is _________ m/s. (Assume string does not slip and $g = 10 \text{ m/s}^2$)
- A spring of force constant 15 N/m is cut into two pieces. If the ratio of their length is 1:3, then the force constant of smaller piece is ______ N/m.
Match the LIST-I with LIST-II
List-I: List-II: A. Magnetic induction I. $[M L T^{-2} A^{-2}]$ B. Magnetic flux II. $[M L^2 T^{-2} A^{-2}]$ C. Magnetic permeability III. $[M L^0 T^{-2} A^{-1}]$ D. Self inductance IV. $[M L^2 T^{-2} A^{-1}]$ Choose the correct answer from the options given below:
