Question

a roller coaster starts at rest from the top of a hill, coasts down, and then does a loop-the-loop of radius 12 mm . if the riders should feel weightless just at the top of the loop, at what height should the hill be? ignore friction. express your answer with the appropriate units.

Answer 1

The height of the hill should be 2.5 times the radius of the loop.

Step-by-step explanation:

To find the height of the hill, we can use the concept of conservation of energy. At the top of the loop, the riders should feel weightless, which means that their apparent weight is zero. This occurs when the normal force is equal to zero. Using the equation for centripetal force, we can set the normal force to zero and solve for the height:

Fc = m * ac = m * v2 / r

At the top of the loop, the apparent weight is the difference between the gravitational force and the centripetal force:

Wapp = mg - Fc

Setting Wapp equal to zero, we can solve for the height:

mg = m * v2 / r

Here, v is the velocity of the roller coaster at the top of the loop. We can find the velocity using the formula for centripetal acceleration:

ac = v2 / r

With ac = g + 1.5g = 2.5g, we can solve for v:

2.5g = v2 / r

Substituting this value of v into the previous equation, we can find the height:

mg = m * (2.5g * r) / r

mg = 2.5mg

Therefore, the height of the hill should be 2.5 times the radius of the loop.