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 calculate 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.
At the top of the Ferris wheel, two forces act on the person: gravity and the normal force. The normal force is the upward force exerted by the platform (or seat) of the Ferris wheel onto the person.
To begin, let's understand the forces acting on the person at the top of the Ferris wheel:
1. Gravity (Fg): This force pulls the person downwards towards the center of the Earth and is equal to the person's real weight (W).
2. Normal force (FN): This force acts perpendicular to the platform or seat of the Ferris wheel and counterbalances the force of gravity. The normal force provides the person's apparent weight (Wa).
At the top of the Ferris wheel, the normal force is directed downwards, while the gravitational force is directed towards the center of the circular path.
Since the Ferris wheel is rotating, there is an additional centripetal force (Fc) acting on the person, which is directed towards the center of the circular path.
For a rotation, the centripetal force is given by Fc = (m * v²) / r, where m is the mass of the person, v is the velocity of the person on the Ferris wheel, and r is the radius of the circular path (half the diameter of the Ferris wheel).
At the top of the Ferris wheel, the magnitude of the normal force (FN) can be found by balancing the forces in the vertical direction:
Fg - FN = 0
Since Fg is the person's real weight (W), the normal force at the top of the Ferris wheel is equal to the person's real weight:
FN = W
Therefore, the apparent weight (Wa) at the top of the Ferris wheel can be found by adding the centripetal force to the real weight:
Wa = FN + Fc
Now, to find the ratio of apparent weight to real weight, we divide the apparent weight by the real weight:
Ratio = Wa / W = (FN + Fc) / W
In this case, we are given the diameter of the Ferris wheel (26.0m) and the time it takes to complete one rotation (15.0s). However, we need additional information such as the mass of the person in order to calculate the centripetal force and ultimately determine the ratio of the person's apparent weight to her real weight at the top of the Ferris wheel.
To find the ratio of a person's apparent weight to her real weight at the top of the Ferris wheel, we can use the concept of centripetal force.
1. First, let's calculate the velocity of a person on the Ferris wheel. The Ferris wheel rotates once every 15.0s, so the time taken for one complete revolution is 15.0s.
- The circumference of the Ferris wheel is given as 26.0m (diameter = 26.0m, so radius = 13.0m).
- The formula for the circumference is C = 2πr, where C is the circumference and r is the radius.
- Plugging in the values, we have C = 2 * π * 13.0m ≈ 81.68m.
- The velocity (v) of a point on the Ferris wheel is given by dividing the circumference by the time for one complete revolution.
- v = C / t = 81.68m / 15.0s ≈ 5.45m/s.
2. Next, we need to calculate the acceleration (a) of a person on the Ferris wheel. The acceleration points towards the center of the circle and is given by a = v^2 / r.
- Plugging in the values, we have a = (5.45m/s)^2 / 13.0m ≈ 2.28m/s^2.
3. Now, let's consider the forces acting on a person on the Ferris wheel at the top. We have the apparent weight (F_apparent) and the real weight (F_real) acting downwards (towards the center of the circle), and the centrifugal force (F_centrifugal) pointing towards the outside of the circle.
- At the top, the apparent weight is the sum of the real weight and the centrifugal force: F_apparent = F_real + F_centrifugal.
- The centrifugal force (F_centrifugal) at the top is given by F_centrifugal = m * a, where m is the mass of the person and a is the acceleration.
- Since the apparent weight is the same as the real weight when the person is at rest, we have F_apparent = m * g, where g is the acceleration due to gravity.
4. The ratio of a person's apparent weight to her real weight at the top is given by the equation:
Ratio = F_apparent / F_real
= (F_real + F_centrifugal) / F_real
Substituting the values we have calculated:
Ratio = (m * g + m * a) / (m * g)
= (g + a) / g.
Therefore, the ratio of a person's apparent weight to her real weight at the top of the Ferris wheel is (g + a) / g, where g is the acceleration due to gravity and a is the acceleration of the person on the Ferris wheel.