A 500 g block on a frictionless surface is attached to a rather limp spring of constant k = 8.7 N/m. A second block rests on the first, and the whole system executes simple harmonic motion with a period of 3.1 s. When the amplitude of the motion is increased to 60 cm, the upper block just begins to slip. What is the coefficient of static friction between the blocks?

To find the coefficient of static friction between the blocks, we need to analyze the forces acting on the upper block when it just begins to slip.

First, let's calculate the angular frequency (ω) of the simple harmonic motion. The period (T) of the motion is given as 3.1 seconds. The formula relating period and angular frequency is T = 2π/ω. Rearranging the formula, we find ω = 2π/T.

ω = 2π / 3.1 s ≈ 2.03 rad/s

Next, we can use the formula for the angular frequency (ω) in terms of the spring constant (k) and mass (m) of the upper block to calculate the mass of the upper block (m). The formula is ω = √(k / m). Rearranging the formula, we find m = k / ω^2.

m = 0.5 kg

Now, let's analyze the forces acting on the upper block when it just begins to slip. At this point, the force of static friction between the blocks is at its maximum. The maximum static friction force (f_smax) can be calculated by multiplying the coefficient of static friction (μ_s) and the normal force (N) between the blocks. Since the blocks are in simple harmonic motion, the extension of the spring (x) is equal to the amplitude of the motion (A).

f_smax = μ_s * N

The normal force (N) can be calculated as the weight of the upper block. The formula for weight is given by N = m * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).

N = 0.5 kg * 9.8 m/s^2 ≈ 4.9 N

Since the amplitude (A) of the motion is given as 60 cm, we must convert it to meters.

A = 60 cm = 0.6 m

Since the upper block only just begins to slip when the amplitude is increased to 60 cm, the static friction force (f_s) is equal to the force exerted by the spring (F_s) at this amplitude.

F_s = k * A

Now, equating the static friction force (f_s) and the force exerted by the spring (F_s), we can solve for the coefficient of static friction (μ_s).

μ_s * N = k * A

μ_s = (k * A) / N

Substituting the known values:

μ_s = (8.7 N/m * 0.6 m) / 4.9 N

μ_s ≈ 1.07

Therefore, the coefficient of static friction between the blocks is approximately 1.07.