The position vectors of two 1 kg particles, (A) and (B), are given by rA=(α1t2i^+α2tj^+α3tk^)m and rB=(β1ti^+β2t2j^+β3tk^)m, respectively; (α1=1 m/s2,α2=3nm/s,α3=2 m/s,β1=2 m/s,β2=−1 m/s2,β3=4pm/s), where t is time, n and p are constants. At t=1 s,VA=VB and velocities VA and VB of the particles are orthogonal to each other. At t=1 s, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is Lkgm2 s−1. The value of L is _______ .
Held on 22 Jan 2025 · Verified 6 Jul 2026.
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Net gravitational force at the center of a square is found to be $F_{1}$ when four particles having mass $M, 2 M, 3 M$ and $4 M$ are placed at the four corners of the square as shown in figure and it is $F_{2}$ when the positions of $3 M$ and $4 M$ are interchanged. The ratio $\frac{F_{1}}{F_{2}}$ is $\frac{\alpha}{\sqrt{5}}$. The value of $\alpha$ is $\_\_\_\_$. 
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The surface tension of a soap solution is $3.5 \times 10^{-2}$ N/m. The work required to increase the radius of a soap bubble from $1$ cm to $2$ cm is $\alpha \times 10^{-6}$ J. The value of $\alpha$ is _____. ($\pi = 22/7$)
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