A simple pendulum consists of a ball of mass 3kg suspended from the ceiling using a string of length 4 meters. The ball is displaced from its equilibrium position by a small angle of 5°. a) What is the magnitude of the restoring force that moves the ball toward its equilibrium position and produces simple harmonic motion?
Question
A simple pendulum consists of a ball of mass 3kg suspended from the ceiling using a string of length 4 meters. The ball is displaced from its equilibrium position by a small angle of 5°. a) What is the magnitude of the restoring force that moves the ball toward its equilibrium position and produces simple harmonic motion?
Solution
To find the magnitude of the restoring force, we first need to understand that for small angles, the restoring force (F) in a simple pendulum is given by the formula:
F = mgsin(θ)
where: m is the mass of the pendulum bob, g is the acceleration due to gravity, and θ is the angle of displacement.
Given in the problem, we have: m = 3 kg, g = 9.8 m/s² (approximate value on the surface of the Earth), and θ = 5°.
However, the trigonometric functions in the formula work with radians, not degrees. So, we first need to convert the angle from degrees to radians. We know that π radians = 180°, so:
θ = 5 * (π / 180) = 0.0873 radians (approx).
Now we can substitute these values into the formula:
F = 3 kg * 9.8 m/s² * sin(0.0873 rad) = 2.54 N.
So, the magnitude of the restoring force is approximately 2.54 Newtons.
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