Question

After a short interval of speeding up from rest, the ride spins at a constant rate about its central axis. Passengers may choose to sit in either an inner seat, located a distance R from the axis, or an outer seat, at a distance 1.7R from the axis. The seats are held in place by very strong, rigid metal bars.Bernice is a bit nervous about going on this ride, because she tends to get motion sickness. However, she agrees to go with Gertrude as long as she can choose a seat with the smallest possible magnitude of acceleration, which should give the most comfortable ride. They are having a discussion about whether the inner seats or the outer seats will be better for this purpose.Bernice: "The radial acceleration is equal to v2/r, so it’s inversely proportional to r. The outer seat is at a larger radius, so it should have less acceleration."Gertrude: "But the radial acceleration is also equal to ω2r, which is directly proportional to r. That would mean the inner seat, at a smaller radius, should have less acceleration."a) Whose argument is correct? Which seat actually has less radial acceleration? What is wrong with the argument that is incorrect? Explain your reasoning.

Answer 1

The trip with less acceleration has it when the radius is smaller, for which Gertrude is right

Step-by-step explanation:

We will propose the solution of this exercise, with the second law of / newton

F = m a

where the acceleration is centripetal, whose equation is

a = v² / r

where v is the linear speed of the chair and r is its radius from the turning point.

Acceleration depends on the two parameters: linear speed and radius, for which reason we must take them into account for an analysis.

angular and linear speeds are related

v = w r

therefore we see that the linear speed also changes with the radius, but the angular speed is constant for the entire apparatus; let's substitute to see which variable is more important

a = (w r)² / r

a = w² r

Since the angular velocity is constant, the relationship dependency is linear with the radius.

The trip with less acceleration has it when the radius is smaller, for which Gertrude is right

Bernice's problem is that it does not take into account that the acceleration depends on the two parameters linear speed and radius