A pendulum has a mass of 3.2 kg, a length of 2 meters and swings through a (half)arc of 28.1 degrees. What is its amplitude to the nearest centimeter?
To find the amplitude of a pendulum, we need to use the formula:
Amplitude = (Length * Arc length) / (2 * pi)
Given:
Mass of the pendulum (m) = 3.2 kg
Length of the pendulum (L) = 2 meters
Arc length (s) = 28.1 degrees
First, we need to convert the arc length from degrees to radians. Since there are 360 degrees in a circle and 2pi radians, we can use the conversion factor:
1 radian = (180 degrees) / pi
Arc length in radians (θ) = (28.1 degrees) * (pi / 180) radians/degree
Next, substitute the given values into the amplitude formula:
Amplitude = (Length * Arc length) / (2 * pi)
Amplitude = (2 meters * θ radians) / (2 * pi)
Amplitude = θ * Length / pi
Now, calculate the amplitude:
Amplitude = [(28.1 degrees) * (pi / 180) radians/degree] * 2 meters / pi
Amplitude ≈ 0.491 meters
Finally, we need to convert the amplitude from meters to centimeters. Since 1 meter is equal to 100 centimeters, we can use the conversion factor:
1 meter = 100 centimeters
Amplitude in centimeters = 0.491 meters * 100 centimeters/meter
Amplitude ≈ 49.1 centimeters
Therefore, the amplitude of the pendulum to the nearest centimeter is 49 centimeters.