To find the point at which the string will tear up, we need to determine the tension in the string at the maximum angular amplitude (60 degrees) of the simple pendulum. At this point, the tension in the string will be at its maximum, and if it exceeds a certain threshold, the string might tear up.
First, let's calculate the tension in the string at the maximum angular amplitude (θ = 60 degrees). The tension (T) in the string at any point during the pendulum's swing can be given by the equation:
T = m * g * (l - l * cos(θ))
where: m = mass of the bob (in kg) = 1 kg g = acceleration due to gravity (approximately 9.8 m/s^2 on Earth's surface) l = length of the pendulum (in meters) = 1 m θ = angular displacement from the vertical (in radians)
However, the equation assumes that the angular displacement is measured from the vertical downward position. In this case, we are given the angular amplitude, which is measured from the vertical upward position. So, we need to adjust the equation accordingly.
The angular displacement from the vertical downward position (θ') can be related to the angular amplitude (θ_amp) as follows:
θ' = π - θ_amp
where θ' is the angular displacement from the vertical downward position and θ_amp is the angular amplitude.
Given that the angular amplitude (θ_amp) is 60 degrees, we can convert it to radians:
θ_amp = 60 degrees * (π / 180 degrees) ≈ 1.0472 radians
Now, we can calculate θ' using the formula:
θ' = π - 1.0472 radians ≈ 2.0944 radians
With θ' calculated, we can now find the tension (T) at the maximum angular amplitude:
T = m * g * (l - l * cos(θ')) T = 1 kg * 9.8 m/s^2 * (1 m - 1 m * cos(2.0944 radians)) T ≈ 1 kg * 9.8 m/s^2 * (1 m + 0.5) T ≈ 14.7 N
The tension in the string at the maximum angular amplitude is approximately 14.7 Newtons. If the string cannot withstand this tension, it might tear up at this point during the pendulum's swing.