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NEET
Exam
NEET
Subject
Physics
Chapter
Mechanical Properties of Fluids

NEET · Physics · Mechanical Properties of Fluids

A tiny spherical glass bead is projected vertically downwards into...

A tiny spherical glass bead is projected vertically downwards into a deep column of highly viscous oil. Its initial downward velocity v0v_0 is twice its terminal velocity (v0=2vtv_0 = 2v_t). Which of the following descriptions correctly shows the variation of its velocity (vv) with time (tt)?

Options

  1. AA curve starting at v0v_0 on the velocity axis and decreasing with a continuously flattening slope to approach a horizontal asymptote at vtv_t.
  2. BA curve starting at v0v_0 on the velocity axis and decreasing continuously to approach zero asymptotically.
  3. CA straight line starting at v0v_0 on the velocity axis and decreasing linearly until it abruptly becomes horizontal at vtv_t.
  4. DA curve starting at v0v_0 on the velocity axis and increasing with a continuously flattening slope to approach a higher horizontal asymptote.

Correct answer

A. A curve starting at v0v_0 on the velocity axis and decreasing with a continuously flattening slope to approach a horizontal asymptote at vtv_t.

Step-by-step solution

The net downward force on the spherical bead is given by Fnet=Weff−FvF_{net} = W_{eff} - F_v, where WeffW_{eff} is the effective weight (weight minus buoyancy)andFvand F_v is the velocity-dependent viscous drag.

At terminal velocity vtv_t, the net force is zero, so Weff=6πηrvtW_{eff} = 6\pi\eta r v_t.

The bead is projected with an initial velocity v0=2vtv_0 = 2v_t. Since v0>vtv_0 > v_t, the initial upward viscous drag is greater than the downward effective weight.

This results in a net upward force, causing the bead to decelerate. As the velocity decreases, the viscous drag decreases, and the net upward force (and thus the retardation) decreases.

The velocity will continuously decrease from v0v_0 and asymptotically approach the terminal velocity vtv_t from above. The graph is a curve starting at v0v_0 and flattening out to a horizontal asymptote at vtv_t.

Answer: A curve starting at v0v_0 on the velocity axis and decreasing with a continuously flattening slope to approach a horizontal asymptote at vtv_t.
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