Explanation

To find the divergence, calculate:

div(f) = ∂/∂x (3x^2) + ∂/∂y (5xy) + ∂/∂z (xyz^3)

= 6x + 5x + 3xyz^2

At (1, 2, 3):

div(f) = 6(1) + 5(1) + 3(1)(2)(3)^2

= 6 + 5 + 54

= 65

Explanation

Bohr's quantum condition states that the angular momentum of an electron in a circular orbit is quantized:

mvr = nh / 2π

where m is the mass of the electron,

v is the velocity,

r is the radius of the orbit,

n is the principal quantum number,

and h is Planck's constant.

Explanation

The acceleration of a particle moving in a uniform circular motion is directed towards the center of the circle and is given by:

a = v^2 / r

This is known as centripetal acceleration.

Explanation

When a spherical ball rolls without slipping:

Total energy (E) = Kinetic energy (K) + Rotational energy (U)

K = (1/2)mv^2

U = (1/2)Iω^2 = (1/5)mv^2 (for a sphere)

E = K + U = (1/2)mv^2 + (1/5)mv^2 = (7/10)mv^2

Fraction of energy associated with rotation:

U/E = ((1/5)mv^2) / ((7/10)mv^2) = 2/7

Explanation

According to the law of universal gravitation:

F ∝ (m1 × m2) / r^2

Initially, F0 ∝ (m × m) / r^2

After doubling the mass and halving the distance:

F_new ∝ (2m × 2m) / (r/2)^2

= 16 × F0

Explanation

We know that for a Gaussian surface, flux = q/ε

Here, q = 10^-6 C and ε = 8.85 X 10^-12 C/Nm^2

Therefore, flux = 10^-6/8.85 X 10^-12 

= 1.12 X 10^5 Nm^2/C.

Explanation

Energy of one photon (E) = hc / λ

h = 6.626 × 10^-34 Js

c = 3 × 10^8 m/s

λ = 4961 Å = 4961 × 10^-10 m

E = (6.626 × 10^-34 Js × 3 × 10^8 m/s) / (4961 × 10^-10 m)

= 4.007 × 10^-19 J

Number of photons required to do 1 J of work:

n = 1 J / (4.007 × 10^-19 J)

= 2.4961 × 10^18

Explanation

Total mechanical energy (E) = (1/2) × m × ω^2 × A^2

ω = 2π / T = 2π / 0.20 s = 10π rad/s

m = 40 g = 0.04 kg

A = 2.0 cm = 0.02 m

E = (1/2) × 0.04 kg × (10π rad/s)^2 × (0.02 m)^2

= 8.0 × 10^-3 J

Explanation

Explanation