how many revolutions per minute would a 5.8 metre radius ferris wheel need to make for the passenger to feel weightless at the topmost point? mass of passenger is 55.00 kg.
The angular velocity of the ferris wheel would need to be 2π radians per second, or 120 revolutions per minute.
To determine the number of revolutions per minute (rpm) needed for a passenger on a ferris wheel to feel weightless at the topmost point, we can use the centripetal force equation.
The formula for centripetal force is F = m * a, where F is the force, m is the mass, and a is the acceleration. In this case, the acceleration is centripetal acceleration, which is given by a = (v^2) / r, where v is the velocity and r is the radius.
When the passenger is at the topmost point of the ferris wheel, the centripetal force required to keep them in circular motion is equal to their weight (mg), where g is the acceleration due to gravity.
So, we have F = mg = m * (v^2) / r
We can rearrange this equation to solve for v:
v^2 = (g * r)
Now, we can calculate v:
v = sqrt(g * r)
Next, we need to determine the time it takes for the ferris wheel to complete one revolution at the topmost point. We know the circumference of a circle is C = 2 * π * r, and since the ferris wheel completes one revolution, which is equivalent to one circumference, the distance travelled is equal to the circumference, C.
Now, let's calculate the time, t, using the formula:
t = C / v
Since the time is usually measured in minutes, we need to convert the time from seconds to minutes.
Finally, we can calculate the number of revolutions per minute (rpm) using the formula:
rpm = 60 / t
Plugging in the values, let's calculate the number of revolutions per minute:
radius (r) = 5.8 meters
mass (m) = 55.00 kg
acceleration due to gravity (g) = 9.8 m/s^2
v = sqrt(g * r) = sqrt(9.8 * 5.8) ≈ 7.64 m/s
C = 2 * π * r = 2 * π * 5.8 ≈ 36.46 meters
t = C / v = 36.46 / 7.64 ≈ 4.77 seconds
Since we want the answer in minutes, we'll convert the time from seconds to minutes:
t_in_minutes = t / 60 = 4.77 / 60 ≈ 0.08 minutes
Finally, we'll calculate the number of revolutions per minute:
rpm = 60 / t_in_minutes = 60 / 0.08 ≈ 750 rpm
Therefore, a 5.8-meter radius ferris wheel would need to make approximately 750 revolutions per minute for the passenger to feel weightless at the topmost point.
To determine the number of revolutions per minute (RPM) needed for the passenger to feel weightless at the topmost point of the ferris wheel, we can use the centripetal force equation and the gravitational force equation.
First, let's calculate the gravitational force acting on the passenger at the topmost point:
Gravitational force (Fg) = mass (m) x acceleration due to gravity (g)
Fg = 55.00 kg x 9.8 m/s^2
Fg = 539 N
At the topmost point of the ferris wheel, the gravitational force acting on the passenger must be equal to the centripetal force, which is given by:
Centripetal force (Fc) = mass (m) x radius (r) x angular velocity squared (ω^2)
Since we want the passenger to feel weightless, the centripetal force must be equal to zero. Therefore, we have:
0 = 55.00 kg x 5.8 m x ω^2
Solving for ω^2:
ω^2 = 0 / (55.00 kg x 5.8 m)
ω^2 = 0 rad^2/s^2
Since ω^2 is zero, it means that the ferris wheel needs to make an infinite number of revolutions per minute for the passenger to feel weightless at the topmost point.
In practical terms, this would mean that the ferris wheel needs to rotate at a speed great enough for the centrifugal force to exactly cancel out the gravitational force on the passenger at the topmost point. This would result in the passenger experiencing zero net force, leading to a feeling of weightlessness.