A swing 5 metres long swings through a vertical angle of 50 degrees. Through what vertical distance does the seat rise in going from its lowest position to its highest?
At the lowest point, the swing is 5m below the support.
At the half-vertical angle of 50, the swing is 5m*cos(25°) below the support.
So the vertical distance
= 5m*(1-cos(25°)
Your calculator will give you the approximation to the exact answer.
To find the vertical distance that the seat of the swing rises from its lowest position to its highest, we can use the concept of arc length.
Arc length is the distance traveled along the circumference of a circle, and it is measured in the same units as the radius of the circle. In this case, the swing is essentially a pendulum, with a length of 5 meters.
The swing swings through an angle of 50 degrees. To find the arc length corresponding to this angle, we need to calculate the circumference of the circle that the swing makes when it swings.
We can use the formula for the circumference of a circle, which is given by:
C = 2πr
In this case, the radius of the circle is the length of the swing, which is 5 meters. So, we have:
C = 2π × 5
To find the arc length corresponding to the angle of 50 degrees, we need to calculate what fraction of the circumference represents this angle. A full circle has an angle of 360 degrees, so the fraction of the circle corresponding to 50 degrees is:
Fraction of circle = 50 degrees / 360 degrees
Now, we can calculate the arc length using the formula:
Arc length = Fraction of circle × Circumference
Plugging in the values, we have:
Arc length = (50 degrees / 360 degrees) × (2π × 5)
Simplifying the expression, we get:
Arc length = (5/36) × (2π × 5)
Finally, we calculate the vertical distance that the seat rises by assuming that the arc length is the same as the vertical distance. Therefore, the vertical distance is:
Vertical distance = Arc length = (5/36) × (2π × 5)
Calculating the value of the expression gives us the answer.