Show that on a roller coaster with a circular vertical loop...

Show that on a roller coaster with a circular vertical loop (Fig. $6-44 )$ , the difference in your apparent weight at the top of the circular loop and the bottom of the circular loop is $6 g^{\prime} s-$ that is, six times your weight. Ignore friction. Show also that as long as your speed is above the minimum needed, this answer doesn't depend on the size of the loop or how fast you go through it. figure can't copy

Key Concepts

Centripetal Acceleration

In circular motion, an object moving along a circular path must continuously accelerate toward the center of the circle. This inward, or centripetal, acceleration arises from the net force acting on the object and is given by the relation a = v²/r, where v is the tangential speed and r is the radius of the circle. This concept is essential when analyzing forces on a roller coaster moving through a loop since the required centripetal force must be provided by the combination of gravitational force and the normal (apparent weight) force.

Circular Motion Dynamics

The analysis of forces in a vertical loop involves decomposing the forces acting on the rider at various points along the loop. At the top of the loop, both gravity and the normal force contribute to the centripetal acceleration (with gravity aiding in providing the necessary inward pull), whereas at the bottom, gravity opposes the centripetal requirement, and the normal force must compensate by being significantly higher. This difference in the distribution of forces is key to understanding the variations in apparent weight experienced by the rider.

Apparent Weight

Apparent weight refers to the normal force experienced by a rider, which is what a scale would read if used to measure weight. On a roller coaster loop, this is not constant because the actual forces on the body change due to the centripetal acceleration required for circular motion. At the bottom of the loop the normal force is significantly higher than the rider’s actual weight (mg), whereas at the top, the normal force is reduced as gravity assists in providing the centripetal force. The difference between these forces creates the sensation of being much heavier or lighter during the ride.

Minimum Speed Condition

For the roller coaster to safely navigate the loop, there is a minimum required speed at the top of the loop that ensures that the normal force does not drop to zero (which would indicate loss of contact between the rider and the track). This constraint is derived from setting the centripetal force equal to the gravitational force at the top of the loop. Only when the ride’s speed exceeds this threshold will the normal force be positive, thereby guaranteeing continuous circular motion without the rider losing contact with the track.

Parameter Independence

A surprising result of the analysis is that, above the minimum speed required to complete the loop, the difference in apparent weight between the top and bottom of the loop becomes a fixed multiple of the actual weight. This outcome, showing a difference of six times the gravitational weight, is independent of both the loop’s size and the precise speed of the coaster, as long as the speed is above the minimum needed. This is a consequence of the way the centripetal acceleration requirements and gravitational force interact at the extreme points of the loop.

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