Question

A Ferris wheel of radius 30 feet is rotating counterclockwise with an angular velocity of 2 radians per second. How fast is a seat on the rim rising (in the vertical direction) when it is 15 feet above the horizontal line through the center of the wheel?

Answer 1

51.96 feet is the value with which the seat on the rim is rising in the vertical direction

Explanation:

Angular velocity = ω= 2 rad/s

radius of ferris wheel = r = 30 feet

We need to find that , how fast is a seat on rim rising (in vertical direction) when it is 15 feet above horizontal line through center of wheel, meaning how quickly a point on rim of wheel is rising vertically at point where this point is 15 ft above horizontal line through center of wheel.

To write the expression :

y = r sin (ω t)

y= 30 sin (2t)

y = 60 sin(t) cos(t)

y= 60
`(cos(t)^(2)- sin (t)^(2) )`

y = 60 cos(2t) _____________________(Equation 1)

The position of 15 feet up is given by angle of 2t

Sin (2t) =
`(15)/(30)` =
`(1)/(2)`

Then,

2t = 30 degrees

cos (2t) = cos (30) =
`(√(3))/(2)`

put in equation 1 , we get:

y = 60cos(30)

y = 60 (0.866)

y= 51.96 feet

which is the value with which the seat on the rim is rising in the vertical direction when it is 15 feet above horizontal line through center of wheel.