Video Transcript
A pendulum of length 26 centimeters
swings 58 degrees. Find the length of the circular
pathway that the pendulum makes giving the answer in centimeters in terms of 𝜋.
In this question, we’re given that
there’s a pendulum, which is 26 centimeters long. This means that the length of
string from the pivot point here at the top to the ball at the end is 26
centimeters. We’re told that the angle that this
pendulum swings through is 58 degrees. And we’re told that it swings in a
circular pathway. We could draw a smaller diagram of
the pendulum, which allows us to say that if this pendulum was to swing the entire
way round, it would in fact create a circle. The length of the string, which is
26 centimeters, would in fact be the radius of the circle.
The length that we need to work out
is marked in green and that’s an arc of the circle. Because we’re given that the
central angle is a measurement in degrees, then we use this formula that the arc
length of a circle of radius 𝑟, with a central angle 𝜃 degrees, is given by arc
length equals two 𝜋𝑟𝜃 over 360. We can remember that this formula
is a result of multiplying the circumference, which is two 𝜋𝑟 by this proportion
of 𝜃 over 360 degrees.
Now all we need to do is plug in
the values that the radius is 26 centimeters and 𝜃, the central angle, is 58
degrees. If we wish, we can take out this
common factor of two before we simplify to give us the answer that the arc length is
377 over 45𝜋 centimeters. In some questions, we might be
asked for a decimal approximation for the length. However, this question asks us for
the length in terms of 𝜋. Therefore, we leave the answer as
it is. So the circular pathway has a
length of 377 over 45 𝜋 centimeters.
