Explanation

Explanation

To find the moment of inertia (I) of the rotating body, we can use the formula:

τ = I × α

where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

Given:

τ = 2 × 10^-4 Nm

α = 4 rad/s^2

Rearranging the formula to solve for I:

I = τ / α

= (2 × 10^-4 Nm) / (4 rad/s^2)

= 0.5 × 10^-4 kgm^2

Explanation

Explanation

To find the time it takes to decay 75% of the initial amount, we need to find the time it takes for 25% of the initial amount to remain.

Since the half-life is 4.5 × 10^9 years, after one half-life, 50% of the initial amount remains.

After two half-lives, 25% of the initial amount remains (50% of 50%).

Therefore, the time it takes to decay 75% of the initial amount is:

2 × half-life = 2 × 4.5 × 10^9 years = 9 × 10^9 years

Explanation

To find the work done, we need to integrate the force with respect to the displacement.

Given:

F(x) = 7 - 2x + 3x^2 N

x_initial = 0 m

x_final = 5 m

Work done (W) is the integral of force (F) with respect to displacement (x):

W = ∫[x_initial, x_final] F(x) dx

= ∫[0, 5] (7 - 2x + 3x^2) dx

Evaluating the integral:

W = [7x - x^2 + x^3] from 0 to 5

= (7(5) - 5^2 + 5^3) - (7(0) - 0^2 + 0^3)

= (35 - 25 + 125) - 0

= 135 J

Explanation

Explanation

To find the smallest Bragg scattering angle, we can use Bragg's law:

2d sin(θ) = nλ

where d is the distance between the atomic planes, θ is the scattering angle, n is an integer (which is 1 for the smallest angle), and λ is the wavelength of the X-ray.

Given:

d = 0.3 nm

λ = 0.03 nm

Rearranging Bragg's law to solve for θ, we get:

sin(θ) = λ / (2d)

= 0.03 nm / (2 × 0.3 nm)

= 0.05

θ = arcsin(0.05)

≈ 2.9°

Explanation

To find the increase in internal energy, we can use the equation:

ΔU = Q - W

where ΔU is the change in internal energy, Q is the heat energy added (which is the latent heat of vaporization in this case), and W is the work done.

Given:

Q = 2240 J (latent heat of vaporization of water per gram)

W = 168 J (work done in the process of vaporization)

Now, we can plug in the values:

ΔU = Q - W

= 2240 J - 168 J

= 2072 J

Explanation

To calculate the moment of inertia of the Earth about its diameter, we can use the formula for the moment of inertia of a sphere:

I = (2/5)MR^2

where:

I = moment of inertia

M = mass of the Earth (10^25 kg)

R = radius of the Earth (diameter/2 = 12800 km / 2 = 6400 km = 6.4 × 10^6 m)

Plugging in the values, we get:

I = (2/5) × 10^25 kg × (6.4 × 10^6 m)^2

= (2/5) × 10^25 kg × 4.096 × 10^13 m^2

= 1.6384 × 10^38 kg m^2

≈ 1.64 × 10^38 kg m^2

Explanation

To calculate the velocity attained by the rocket, we can use the equation:

F × t = m × Δv

where:

F = force applied (5 × 10^5 N)

t = time (20 s)

m = mass of the rocket (2 × 10^4 kg)

Δv = change in velocity (final velocity - initial velocity)

Since the rocket starts from rest, the initial velocity is 0. Therefore, Δv = final velocity.

Rearranging the equation to solve for Δv, we get:

Δv = F × t / m

= (5 × 10^5 N) × (20 s) / (2 × 10^4 kg)

= 500 m/s