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Questions and Answers

What is the angular velocity of the frog in the COM frame?

  • $\sqrt{\frac{2v_0}{m}}$
  • $\frac{v_0}{2m}$
  • $\frac{2v_0}{m}$
  • $\frac{v_0}{m}$ (correct)

What is the velocity of block B with respect to the COM frame?

  • $v_0 cos(\omega t) + v_0$
  • $-v_0 + v_0 cos(\omega t)$ (correct)
  • $v_0 + v_0 cos(\omega t)$
  • $v_0 cos(\omega t) - v_0$

What is the range of the frog in the ground frame?

  • $\frac{2v_0^2}{g}$
  • $\frac{6v_0^2}{g}$
  • $\frac{3v_0^2}{g}$
  • $\frac{12v_0^2}{g}$ (correct)

What is the time of flight of the frog in the ground frame?

<p>$\frac{6v_0}{g}$ (C)</p>
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What is the initial velocity of the frog in the COM frame?

<p>$v_0$ (B)</p>
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What is the displacement of the frog in the COM frame?

<p>$\lambda_0 + v_0T - v_0 sin(\omega T)$ (B)</p>
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What is the expression for λ0?

<p>$R - v_0T + v_0 sin(\omega T)$ (A)</p>
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What is the final expression for λ0?

<p>$\frac{6v_0^2}{g} - v_0 \pi$ (D)</p>
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What is the condition for the angular momentum of a system to be conserved?

<p>The net torque acting on the system is zero (A)</p>
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What is the expression for the angular velocity ω after the collision?

<p>v0/(3R) (C)</p>
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What is the condition for the mechanical energy to be conserved after the collision?

<p>The system is elastic (B)</p>
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What is the value of v0 in terms of g and h?

<p>2gh (A)</p>
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What is the minimum condition for the projectile to graze the cylinder?

<p>It touches the cylinder at two points (D)</p>
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What is the value of ω in terms of R and g?

<p>√(g/(3R)) (C)</p>
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What is the change in potential energy equal to?

<p>2x^2 - 2x + 16x (D)</p>
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What is the value of K at 298 K?

<p>2.303 log (50/100) (B)</p>
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What is the possible transition of 3 and 4?

<p>3 → 2 and 4 → 2 (C)</p>
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What is the molecular mass of A2B4?

<p>100 (D)</p>
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What is the ratio of A to B?

<p>1:2 (A)</p>
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What is the value of α at 10 min?

<p>50% (D)</p>
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What is the percentage of B2 at 10 min?

<p>50% (D)</p>
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What is the percentage of A2 at 10 min?

<p>25% (D)</p>
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What is the expression for u in terms of v and g?

<p>u = v + 2gR(1 + cos α) (D)</p>
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What is the angle α for which u is minimum?

<p>45° (A)</p>
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What is the relationship between u and v at the instant just after the jump?

<p>u = 5v (B)</p>
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What is the expression for u^2 in terms of R and g?

<p>u^2 = Rg(2 + 2cos^2 α) (B)</p>
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What type of bond is present in AlCl3?

<p>(3C – 4e) bond (B)</p>
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What is the value of tan^2 θ0 in terms of R and g?

<p>tan^2 θ0 = 3 + 2^2 (C)</p>
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What is the expression for v^2 in terms of R and g?

<p>v^2 = 2Rg sin α (C)</p>
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How many (3C – 2e) bonds are present in B2H6?

<p>Two (B)</p>
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What is the condition for orbitals to be degenerate in an electron system?

<p>In a one-electron system (C)</p>
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What is the minimum value of u^2 in terms of R and g?

<p>u^2 = 4Rg (B)</p>
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What is the relationship between u and v just before the jump?

<p>u = v (B)</p>
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Which orbitals are degenerate in the second excited state?

<p>Only three p-orbitals (2p) (A)</p>
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What is the value of x in Sol.x = 9, y = 3, z = 2?

<p>9 (A)</p>
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What is the number of (3C – 2e) bonds in the given molecule?

<p>6 (B)</p>
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Which element is present in the molecule BeH2?

<p>Beryllium (C)</p>
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How many (3C – 2e) bonds are present in a molecule with 4-(3C-2e) bonds?

<p>4 (B)</p>
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Study Notes

Conservation of Angular Momentum

  • Angular momentum of a system is conserved about the axis passing through the pivots.
  • Li (just before collision) = Lf (just after collision)
  • mv0R = (mR2 + (5m)R2)ω

Conservation of Mechanical Energy

  • Applying conservation of mechanical energy after the collision:
    • 1/2[mR2 + (5m)R2]ω2 + mgR = 5mg
    • ω = 3R/8
    • v0 = 2gh → h = 21R/8
    • ω = √(7g/12R)

Minimum Velocity

  • For u to be minimum, it just grazes the cylinder at two points.
  • u2 = v2 + 2gh
  • u2 = Rg(2 + 2cosα)
  • For u min, dα/d(u2) = 0 → cosα = 1/2
  • α = 45°
  • u02 = Rg(2 + 2)

Collision and Motion

  • After the jump by the frog, v = 2v0
  • vBC = vB - vCM = v0cos(ωt)
  • ω = √(2k/m)
  • vB = vCM + v0cos(ωt)

Projectile Motion

  • R = 2(3v0/g)
  • T = 6v0/g
  • λ0 = R - ∫vBdt
  • λ0 = R + v0T - v0sin(ωT)
  • λ0 = R + v0T - v0sin(6v0/g)

Potential Energy

  • Change in potential energy = - mg(Rsinθ - 2x) + 2x
  • ΔPE = -14x/8 + 7x/4

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