In this pendulum problem, we analyze a mass of 100 grams attached to a 1-meter long string, initially displaced to 7 degrees from its equilibrium position. The goal is to determine the time it takes for the pendulum to swing to a position where it reaches -4 degrees on the opposite side.

To begin, we establish the parameters of the pendulum: the mass (m) is 0.1 kg (converted from grams), the length (L) is 1 meter, the maximum angle (θ_max) is 7 degrees, and the final angle (θ_final) is -4 degrees. It is crucial to convert these angles from degrees to radians for calculations, using the conversion factor π/180. Thus, θ_max becomes approximately 0.12 radians, and θ_final becomes approximately -0.07 radians.

We utilize the equation for angular displacement in simple harmonic motion, which is given by:

θ(t) = θ_max * cos(ωt)

Here, θ(t) represents the angle at time t, and ω is the angular frequency. Rearranging the equation to solve for time (t), we have:

cos(ωt) = θ(t) / θ_max

Substituting the known values:

cos(ωt) = -0.07 / 0.12

Calculating this gives us cos(ωt) ≈ -0.58. To isolate ωt, we apply the inverse cosine function:

ωt = cos^-1(-0.58)

This results in ωt ≈ 2.19 radians.

Next, we need to determine the angular frequency (ω). For a simple pendulum, ω can be calculated using the formula:

ω = √(g / L)

Where g is the acceleration due to gravity (approximately 9.8 m/s²) and L is the length of the pendulum (1 meter). Thus:

ω = √(9.8 / 1) ≈ 3.13 radians/second.

Now, substituting ω back into the equation for t:

2.19 = 3.13t

Solving for t gives:

t = 2.19 / 3.13 ≈ 0.7 seconds.

This result indicates that it takes approximately 0.7 seconds for the pendulum to swing from 7 degrees to -4 degrees on the opposite side. Understanding the principles of pendulum motion, including angular displacement and frequency, is essential for solving similar problems in physics.

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