A solid sphere of mass `m = 2 kg` and specific gravity `s = 0.5` is held stationary relative to a tank filled with water. The tank is accelerating upward with acceleration `2 m//s^2`. Calculate
(a) Tension in the thread connected between the sphere and the bottom of the tank.
(b) If the thread snaps, calculate the acceleration of sphere with respect to the tank.
(Density of water `= 1000 kg m^-3, g = 10 ms^-2`).
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Explanation

In accelerating system, upthrust `F_B = rho Vg_(eff)`
Density of sphere = specific gravity `xx rho_omega = 0.5 xx 1000`
=`500 kg//m^3`
The forces on sphere
`F_B = rho_omega V_(g_(eff)) = rho_omega (m)/(sigma) g_(eff)`
`F_B = rho_omega Vg = rho_omega (m)/(sigma) g_(eff)`
`F_B - T - mg = ma`
`T = F_B - mg - ma = (rho_omega)/(sigma) mg_(eff) - m(g + a)`
=`(rho_omega)/(sigma) m(g + a) -m(g +a)`
=`m(g +a)((rho_omega)/(sigma) -1)`
=`2(10 -2)((1000)/(500) -1) = 24 N`
(b) `T = 0`
`F_B - mg = ma'`
`(rho_omega)/(sigma)m(g + a) - m(g+a)`
`(1000)/(500) xx 2 (10 + 2) -2 xx 10 = 2a'`
`48 - 20 = 2a'`
`a' = 14 m//s^2`
`a_(b//t) = a' - a = 14 - 2 = 12 m//s^2`.


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