a pendulum length 12.3 inches swings 2 degrees and 57 minutes to each side of its vertical position. To the nearest hundredth of an inch, what is the lenght of the arc through which the end of the pendumlum swings?
radius, r = 12.3"
θ
=2*(2°57')
=2*(2+57/60)°*π/180 radians
=0.10297 radian
Arc length = rθ
=1.27"
To find the length of the arc through which the end of the pendulum swings, we can use the formula for the length of an arc of a circle:
Arc length = θ * r
where θ is the central angle in radians, and r is the radius of the circle.
Given that the pendulum swings 2 degrees and 57 minutes to each side of its vertical position, we need to convert this angle to radians.
1 degree is equal to π/180 radians.
57 minutes is equal to 57/60 = 0.95 degrees.
So, the total central angle in degrees is 2 + 0.95 + 2 + 0.95 = 5.9 degrees.
Converting this to radians:
θ = 5.9 * π/180
Now, we need to find the radius of the circular path of the pendulum. The radius is equal to the length of the pendulum.
Given that the length of the pendulum is 12.3 inches, the radius is also 12.3 inches.
Plugging in the values into the formula:
Arc length = (5.9 * π/180) * 12.3
Using the value of π as 3.14 and rounding to the nearest hundredth:
Arc length ≈ (5.9 * 3.14/180) * 12.3 ≈ 0.102 inches
Therefore, to the nearest hundredth of an inch, the length of the arc through which the end of the pendulum swings is approximately 0.10 inches.
To find the length of the arc through which the end of the pendulum swings, we can use the formula for the length of a circular arc, which is given by:
Arc Length = (θ/360) * 2π * r
Where:
- θ is the angle in degrees
- r is the radius of the circle
In this case, the angle is 2 degrees and 57 minutes. To convert minutes to degrees, we divide the number of minutes by 60:
2 degrees + (57 minutes / 60 minutes) = 2.95 degrees (rounded to the nearest hundredth)
The radius of the pendulum is given as 12.3 inches.
Now we can substitute these values into the formula:
Arc Length = (2.95/360) * 2π * 12.3
To calculate this, we'll use the value of π as approximately 3.14159:
Arc Length = (2.95/360) * 2 * 3.14159 * 12.3
Arc Length = (0.008194444) * 6.28318 * 12.3
Arc Length ≈ 0.06455 * 12.3
Arc Length ≈ 0.7946 inches (rounded to the nearest hundredth)
Therefore, the length of the arc through which the end of the pendulum swings is approximately 0.79 inches.