The pendulum of a grandfather clock is 90cm long, when the pendulum swings from one side to the other it travels a horizontal distance of 9.2cm, determine the angle in which the pendulum swings.
tan(θ/2) = 4.6/90
To determine the angle in which the pendulum swings, we can use the concept of arc length and the formula for the length of the arc of a circle.
The length of the arc is given by the formula:
Arc Length = Angle (in radians) × Radius
In this case, the "radius" is the length of the pendulum (90 cm), and the "arc length" is the horizontal distance the pendulum travels (9.2 cm).
Therefore, we can rearrange the formula:
Angle (in radians) = Arc Length / Radius
Angle (in radians) = 9.2 cm / 90 cm
Now we can calculate the angle using the given values:
Angle (in radians) = 0.102 radians
To convert this angle to degrees, we can use the fact that there are 180 degrees in pi radians.
Angle (in degrees) = Angle (in radians) × (180 degrees / pi radians)
Angle (in degrees) = 0.102 radians × (180 degrees / pi radians)
Angle (in degrees) ≈ 5.84 degrees
Therefore, the angle in which the pendulum swings is approximately 5.84 degrees.
To determine the angle in which the pendulum swings, we can use the concept of simple harmonic motion.
In simple harmonic motion, the angle through which the pendulum swings (measured in radians) is directly proportional to the horizontal distance it travels.
The formula for the relationship between the angle (θ), the length of the pendulum (L), and the horizontal distance (d) is given by:
θ = asin(d / L)
where "asin" is the inverse sine function.
In this case, the length of the pendulum (L) is given as 90 cm and the horizontal distance (d) is given as 9.2 cm.
Substituting these values into the formula, we have:
θ = asin(9.2 / 90)
Now we can calculate the value of the angle by using a calculator or a mathematical software that supports trigonometric functions, such as the arcsine (asin) function.
Using a calculator, we find:
θ ≈ 0.1046 radians
Therefore, the angle in which the pendulum swings is approximately 0.1046 radians.