A roller coaster includes a circular vertical loop with radius 16.0 m. For what linear speed at the top of the loop will the passengers feel weightless?

To solve this problem, we need to use the concept of centripetal force. When the passengers feel weightless at the top of the loop, the gravitational force acting on them is providing the necessary centripetal force for the circular motion of the roller coaster.

The formula for centripetal force is Fc = mv^2/r, where m is the mass, v is the velocity, and r is the radius. The gravitational force is Fg = mg, where g is the acceleration due to gravity.

At the top of the loop, these two forces are equal because the gravitational force is providing the necessary centripetal force. So, we can set the two equations equal to each other:

mv^2/r = mg

We can cancel out the mass on both sides, which leaves us with:

v^2/r = g

We can solve for v by multiplying both sides by r and then taking the square root:

v = sqrt(gr)

Substituting the given values (r = 16.0 m, g = 9.8 m/s^2), we get:

v = sqrt(16.0 m * 9.8 m/s^2) = 16 m/s

So, the passengers will feel weightless at the top of the loop if the roller coaster is moving at a speed of 16 m/s.