Suppose that after you are loaded into a Ferris wheel car, the wheel begins turning at 3 revolutions per second. The wheel has diameter 14 meters and the bottom seat of the wheel is 1 meter above the ground. Express the height h of the seat above the ground as a function of time t seconds after it begins turning.
since cos(t) starts at the top, and we are starting at the bottom (I assume, though it is not explicitly indicated),
h(t) = -cos(kt)
the wheel's radius is 7, and the center is 8m up, so
h(t) = 8-7cos(kt)
The period is 1/3, so 2π/k = 1/3, and k=6π
h(t) = 8-7cos(6πt)
To express the height h of the seat above the ground as a function of time t seconds after the Ferris wheel begins turning, we can use the equation of motion for circular motion. The equation is given by:
h = r + r * cos(θ)
Where:
h is the height of the seat above the ground,
r is the radius of the Ferris wheel (half the diameter),
θ is the angle of rotation.
Given that the diameter of the Ferris wheel is 14 meters, the radius r is half of 14, so r = 7 meters.
Since the wheel is turning at a rate of 3 revolutions per second, which means it completes 3 * 2π radians in one second, the angular speed ω is 3 * 2π rad/s.
The equation for the angle θ as a function of time is then:
θ = ω * t
Substituting the values, we have:
θ = (3 * 2π) * t
= 6πt
Now, substituting θ into the equation for h:
h = 7 + 7 * cos(6πt)
Therefore, the height h of the seat above the ground as a function of time t seconds after the Ferris wheel begins turning is h = 7 + 7 * cos(6πt).
To express the height of the seat above the ground as a function of time, let's assume that the initial position of the seat (at t = 0) is at the bottom, 1 meter above the ground.
The Ferris wheel completes a full revolution every 1/3 seconds since it's turning at 3 revolutions per second. Since the height of the seat will repeat every rotation, we can consider one complete revolution of the Ferris wheel, which corresponds to 2π radians.
When the seat is at the bottommost position, it is at a height of 1 meter above the ground. As the wheel rotates, the height of the seat will change due to the circular motion.
The height of the seat can be determined using the equation for vertical motion in circular motion, given by:
h = r + R * cos(θ)
Where:
- h is the height of the seat above the ground
- r is the radius of the wheel, which is half the diameter (7 meters)
- R is the maximum displacement from the center (radius + height of the seat above the ground)
- θ is the angle in radians
Since we want to express the height as a function of time, we can relate the angle θ to time t using the angular velocity, which is the rate at which the angle changes with respect to time.
The angular velocity can be calculated by dividing the number of revolutions per second by the total angle completed in one revolution (2π radians). In this case, the angular velocity (ω) is given by:
ω = 3 * 2π = 6π radians per second
Thus, the angle θ can be calculated as θ = ω * t.
Substituting the values of r, R, and θ into the equation for vertical motion, we get:
h = 7 + (1 + 7) * cos(6π * t)
Simplifying this expression, we have:
h = 8 + 8 * cos(6π * t)
Therefore, the height h of the seat above the ground can be expressed as a function of time t seconds after it begins turning as h = 8 + 8 * cos(6π * t).