A Ferris wheel 26.0m in diameter rotates once every 15.0s What is the ratio of a person's apparent weight to her real weight at the top?

To find the ratio of a person's apparent weight to her real weight at the top of the Ferris wheel, we need to consider the centripetal force acting on the person.

The centripetal force is given by the formula:
Fc = m * ω² * r

Where:
Fc = centripetal force
m = mass of the person
ω = angular velocity (rate of rotation in radians per second)
r = radius of the Ferris wheel

We can find the angular velocity using the equation:
ω = 2π / T

Where:
T = time period for one complete rotation

Given:
Diameter of the Ferris wheel = 26.0 m
Radius of the Ferris wheel (r) = Diameter / 2 = 26.0 m / 2 = 13.0 m
Time period for one rotation (T) = 15.0 s

Calculating the angular velocity:
ω = 2π / T
= 2π / 15.0 s
≈ 0.419 radians/s

Now we can calculate the ratio of apparent weight to real weight at the top of the Ferris wheel.

At the top of the Ferris wheel, the centripetal force acting on the person is the difference between the apparent weight and the actual weight. The apparent weight is the sum of the real weight and the centripetal force.

Real weight (W) = m * g
Where:
g = acceleration due to gravity = 9.8 m/s²

Centripetal force (Fc) = m * ω² * r

Apparent weight (Wa) = W + Fc

Therefore,
Wa = m * g + m * ω² * r

The ratio of apparent weight to real weight (Wa/W) can be calculated as:
(Wa/W) = (m * g + m * ω² * r) / (m * g)

Simplifying the equation, we find:
(Wa/W) = 1 + (ω² * r / g)

Now we can substitute the given values to find the ratio:

(Wa/W) = 1 + (0.419² * 13.0 / 9.8)
(Wa/W) = 1 + (0.175861 * 13.0 / 9.8)
(Wa/W) = 1 + (0.175861 * 1.32653061224)

Calculating the value:
(Wa/W) ≈ 1 + 0.2329
(Wa/W) ≈ 1.2329

Therefore, the ratio of a person's apparent weight to her real weight at the top of the Ferris wheel is approximately 1.2329.

To find the ratio of a person's apparent weight to her real weight at the top of the Ferris wheel, we need to consider the forces acting on the person.

The apparent weight of a person is the force exerted on them by the surface they are standing on, whereas the real weight is the force due to gravity, which is constant regardless of the person's position.

At the top of the Ferris wheel, the person experiences two forces: the downward force due to gravity (real weight) and the upward force due to the circular motion of the Ferris wheel. This upward force is responsible for giving the person an apparent weight.

To calculate the ratio of the person's apparent weight to her real weight, we can use the centripetal force equation:

F_c = m * a_c,

where F_c is the centripetal force, m is the mass of the person, and a_c is the centripetal acceleration.

The centripetal force is provided by the net force acting towards the center of the circular motion, which in this case is the difference between the apparent weight and the real weight:

F_c = F_apparent - F_gravity.

At the top of the Ferris wheel, the net force towards the center is equal to the centripetal force:

F_net = F_c = m * a_c.

Since the person is not accelerating radially (moving in a straight line), the centripetal acceleration can be calculated using the formula:

a_c = v^2 / r,

where v is the linear velocity and r is the radius (half the diameter) of the Ferris wheel.

We can find the linear velocity using the formula:

v = 2 * pi * r / T,

where T is the time taken for one complete revolution of the Ferris wheel.

Now, let's plug in the given values:

Diameter of the Ferris wheel (D) = 26.0m,
Radius of the Ferris wheel (r) = D / 2 = 26.0m / 2 = 13.0m,
Time for one complete revolution (T) = 15.0s.

Using the formulas:

v = 2 * pi * r / T = 2 * pi * 13.0m / 15.0s = 8.69m/s,

a_c = v^2 / r = (8.69m/s)^2 / 13.0m = 5.79m/s^2.

Now that we have the centripetal acceleration, we can calculate the apparent weight of the person using the formula:

F_c = m * a_c.

Since the apparent weight is the net force (F_c) at the top of the Ferris wheel, we can rearrange the equation to find the ratio of the apparent weight to the real weight:

F_apparent / F_gravity = F_c / F_gravity = (m * a_c) / (m * g),

where g is the acceleration due to gravity (9.8 m/s^2).

Finally, we can substitute the values and calculate the ratio:

F_apparent / F_gravity = (m * a_c) / (m * g) = a_c / g = 5.79m/s^2 / 9.8m/s^2 ≈ 0.59.

Therefore, the ratio of a person's apparent weight to her real weight at the top of the Ferris wheel is approximately 0.59.