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 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

In a vacuum or free space, there are no particles or matter, so:

• Pressure (p) is zero.
• Current density (J) is also zero, since there are no charged particles to carry current.

Therefore, the correct observation is p = 0, J = 0.

Explanation

Explanation

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 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

Since μ = 1/√3, we can use the fact that μ = tan(θ) when the block just slides down the inclined plane.

μ = 1/√3 = tan(θ)

θ = arctan(1/√3)

θ = 30°

Explanation

When a satellite orbits close to the Earth's surface, it must travel faster to avoid falling towards the Earth due to gravity. This means its speed is maximum.

Since kinetic energy (KE) is proportional to the square of the speed (v), the kinetic energy will also be maximum:

KE = (1/2)mv^2

where m is the mass of the satellite.

So, when a satellite orbits close to the Earth's surface, its kinetic energy will be maximum.