A person with a mass of 76 kg is moving at a constant speed in a vertical circle on a Ferris wheel with a radius of 37.6 meters. If the difference between his normal force at the bottom (FNB) and the normal force at the top (FNT) of the circle is one third of his weight (FNB-FNT = 1/3 w), with what speed must the Ferris wheel be rotating in m/s? (What's the person's speed another words?)
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To find the person's speed on the Ferris wheel, we can use the concept of centripetal force. In a circular motion like this, the centripetal force is responsible for keeping the object moving in a circle.
In this case, the centripetal force is provided by the combination of the person's weight (mg) acting downwards and the normal force (FNB and FNT) acting towards the center of the circle.
The net force acting towards the center of the circle can be found using the equation:
net force = centripetal force = (m * v^2) / r, where m is the mass of the person, v is the speed, and r is the radius of the circle.
Since we are given that the difference between the normal forces at the bottom and the top is one third of the person's weight, we can write:
FNB - FNT = 1/3 * mg
Since FNB = mg + FNT (because the normal forces add up to the weight), we can substitute this into the equation above:
mg + FNT - FNT = 1/3 * mg
Simplifying the equation, we get:
2/3 * mg = 1/3 * mg
Solving for the speed (v), we can rearrange the equation:
(m * v^2) / r = 2/3 * mg
v^2 = (2/3) * r * g
v = sqrt((2/3) * r * g)
Now, we can substitute the given values into the equation to calculate the speed:
v = sqrt((2/3) * 37.6 * 9.8)
v ≈ 10.44 m/s
Therefore, the person's speed on the Ferris wheel should be approximately 10.44 m/s.