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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?

$\frac{6v_0}{g}$ Signup and view all the answers

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

$v_0$ Signup and view all the answers

What is the displacement of the frog in the COM frame?

$\lambda_0 + v_0T - v_0 sin(\omega T)$ Signup and view all the answers

What is the expression for λ0?

$R - v_0T + v_0 sin(\omega T)$ Signup and view all the answers

What is the final expression for λ0?

$\frac{6v_0^2}{g} - v_0 \pi$ Signup and view all the answers

What is the condition for the angular momentum of a system to be conserved?

The net torque acting on the system is zero Signup and view all the answers

What is the expression for the angular velocity ω after the collision?

v0/(3R) Signup and view all the answers

What is the condition for the mechanical energy to be conserved after the collision?

The system is elastic Signup and view all the answers

What is the value of v0 in terms of g and h?

2gh Signup and view all the answers

What is the minimum condition for the projectile to graze the cylinder?

It touches the cylinder at two points Signup and view all the answers

What is the value of ω in terms of R and g?

√(g/(3R)) Signup and view all the answers

What is the change in potential energy equal to?

2x^2 - 2x + 16x Signup and view all the answers

What is the value of K at 298 K?

2.303 log (50/100) Signup and view all the answers

What is the possible transition of 3 and 4?

3 → 2 and 4 → 2 Signup and view all the answers

What is the molecular mass of A2B4?

100 Signup and view all the answers

What is the ratio of A to B?

1:2 Signup and view all the answers

What is the value of α at 10 min?

50% Signup and view all the answers

What is the percentage of B2 at 10 min?

50% Signup and view all the answers

What is the percentage of A2 at 10 min?

25% Signup and view all the answers

What is the expression for u in terms of v and g?

u = v + 2gR(1 + cos α) Signup and view all the answers

What is the angle α for which u is minimum?

45° Signup and view all the answers

What is the relationship between u and v at the instant just after the jump?

u = 5v Signup and view all the answers

What is the expression for u^2 in terms of R and g?

u^2 = Rg(2 + 2cos^2 α) Signup and view all the answers

What type of bond is present in AlCl3?

(3C – 4e) bond Signup and view all the answers

What is the value of tan^2 θ0 in terms of R and g?

tan^2 θ0 = 3 + 2^2 Signup and view all the answers

What is the expression for v^2 in terms of R and g?

v^2 = 2Rg sin α Signup and view all the answers

How many (3C – 2e) bonds are present in B2H6?

Two Signup and view all the answers

What is the condition for orbitals to be degenerate in an electron system?

In a one-electron system Signup and view all the answers

What is the minimum value of u^2 in terms of R and g?

u^2 = 4Rg Signup and view all the answers

What is the relationship between u and v just before the jump?

u = v Signup and view all the answers

Which orbitals are degenerate in the second excited state?

Only three p-orbitals (2p) Signup and view all the answers

What is the value of x in Sol.x = 9, y = 3, z = 2?

9 Signup and view all the answers

What is the number of (3C – 2e) bonds in the given molecule?

6 Signup and view all the answers

Which element is present in the molecule BeH2?

Beryllium Signup and view all the answers

How many (3C – 2e) bonds are present in a molecule with 4-(3C-2e) bonds?

4 Signup and view all the answers

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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