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A particle is constrained to move on the frictionless inner surface of a parabolic bowl whose cross-section is described by z = k*r². The particle starts at height z0 above the bowl's lowest point with horizontal velocity v0. Gravitational acceleration is g. Which of the following statements are correct? (A) The particle moves in a horizontal circle when v0 = sqrt(g*z0). (B) For v0 > sqrt(2*g*z0), the maximum height reached by the particle is v0² / (2*g). (C) If v0 = 0 and z0 is very small, the time for the particle to return to its starting point is 4*sqrt(2*z0/g). (D) If v0 = 0 and z0 is very small, the time for the particle to return to its starting point is pi*sqrt(2) / sqrt(k*g).

1. (A) The particle moves in a horizontal circle when v0 = sqrt(g*z0).
2. (B) For v0 > sqrt(2*g*z0), the maximum height reached by the particle is v0² / (2*g).
3. (C) If v0 = 0 and z0 is very small then the time in which the particle returns to its initial point is 4*sqrt(2*z0/g).
4. (D) If v0 = 0 and z0 is very small, then the time in which the particle returns to its initial point is pi*sqrt(2) / sqrt(k*g).

Correct answer: (A) The particle moves in a horizontal circle when v0 = sqrt(g*z0).

Solution

For circular orbit at height z0: tan(alpha) = dz/dr = 2*k*r where r = sqrt(z0/k). Normal force components: N*sin(alpha) = m*v0²/r (centripetal), N*cos(alpha) = mg. So tan(alpha)=v0²/(rg). Also tan(alpha)=2*k*r=2*sqrt(k*z0). Thus v0² = 2*k*r²*g = 2*g*z0. So v0 = sqrt(2*g*z0) for circular orbit. Option A says v0=sqrt(g*z0) which is wrong by factor sqrt(2). For option D: small oscillation frequency omega = sqrt(2*k*g), period T = 2*pi/sqrt(2*k*g) = pi*sqrt(2)/sqrt(k*g). D is correct. B: energy conservation: (1/2)*m*v0² + m*g*z0 = m*g*z_max (if no centrifugal term, but angular momentum is conserved so minimum r is not zero). For v0>sqrt(2gz0), angular momentum L=m*v0*r0 is nonzero so particle doesn't reach axis; z_max by energy is (v0²/(2g)+z0) but the true max might differ. Actually for v0>0 the particle has angular momentum so it doesn't go through the axis; z_max via energy = z0 + v0²/(2g) which matches option B statement v0²/(2g) only if z0 term is ignored — so B appears incorrect unless they measure height from bottom differently. Most standard solutions to this problem confirm A is wrong and D is correct.

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