At a height equal to the radius of the Earth, the gravitational force becomes one-fourth of its value at the surface.

Taking the cube root of both sides, a = 27^(1/3) = 3.

Since 27 = 3^3, we have a = 3.

Substituting a = b^k into a^x = b^y gives (b^k)^x = b^y, leading to kx = y, hence x = ky.

From a^x = b^y, taking log_a on both sides gives log_a(b^y) = x, hence y * log_a(b) = x, thus log_a(b) = y/x.

Using the grating equation d sin(θ) = mλ, where d = 1/1000 mm = 1 x 10^-6 m, for m=1, we find θ ≈ 30 degrees.

Since 1000 = 10^3, log_10(1000) = 3, thus x = 3.

The variable resistor allows for adjustment to achieve balance in the bridge.

Using the balance condition R1/R2 = R3/R4, we find R4 = (R2 * R3) / R1 = (15 * 5) / 10 = 7.5Ω.

For balance, R1/R2 = R3/R4, thus R4 = (R2 * R3) / R1 = (20 * 30) / 10 = 60Ω.

The bridge is not balanced because 10/20 ≠ 30/60.

Using the balance condition R1/R2 = R3/R4, we have 2/3 = 6/R4. Solving gives R4 = 4Ω.

Using the balance condition R1/R2 = R3/R4, we have 2/3 = 6/R4. Solving gives R4 = 4Ω.

Using the balance condition R1/R2 = R3/R4, we find R4 = (R2 * R3) / R1 = (10 * 15) / 5 = 30Ω.

For balance, R4 must be 30Ω since 5/15 = 10/30.

Using the formula A(t) = A_0 e^(-ζω_nt), we find ζ = 0.2.

Maximum velocity V_max = Aω. If A is doubled, V_max also doubles.

The total energy E in SHM is given by E = (1/2)kA². If A is doubled, E becomes (1/2)k(2A)² = 4(1/2)kA², which is quadrupled.

Maximum velocity V_max = ωA. If A is halved, V_max is also halved.

The total energy in simple harmonic motion is proportional to the square of the amplitude. If amplitude is doubled, energy increases by a factor of 2^2 = 4.

The total energy of a simple harmonic oscillator is proportional to the square of the amplitude. If the amplitude is doubled, the energy increases by a factor of 2^2 = 4.

The total energy in SHM is proportional to the square of the amplitude. If amplitude is halved, energy is reduced to (1/2)^2 = 1/4, which is halved.

The energy of a wave is proportional to the square of the amplitude. If the amplitude is doubled, the energy increases by a factor of 2^2 = 4.

The energy of a wave is proportional to the square of its amplitude. Therefore, if the amplitude is doubled, the energy increases by a factor of four.

Energy is proportional to the square of the amplitude, so if amplitude is tripled, energy increases by a factor of 3^2 = 9.

The energy of a wave is proportional to the square of its amplitude, so if amplitude is tripled, energy increases by a factor of 3^2 = 9.

If the angle is 90 degrees, the sine of the angle is zero, resulting in zero contribution to the magnetic field from that current element.

Torque is maximum when the angle is 90 degrees because τ = F × r × sin(θ) and sin(90°) = 1.

Torque (τ) = F × d × sin(θ) = 15 N × 2 m × sin(90°) = 30 Nm.

When the angle between the transmission axes of two polarizers is 90 degrees, no light passes through, resulting in zero intensity.