Riders on the "Great American Revolution" loop-the-loop...

Riders on the "Great American Revolution" loop-the-loop roller coaster of Example $5.7$ wear seatbelts as the roller coaster negotiates its $6.7-\mathrm{m}$-radius loop at $9.5 \mathrm{~m} / \mathrm{s}$. At the top of the loop, what are the magnitude and direction of the force exerted on a $55-\mathrm{kg}$ rider (a) by the roller-coaster seat and (b) by the seatbelt? (c) What would happen if the rider unbuckled at this point?

Riders on the "Great American Revolution" loop-the-loop roller coaster of Example $5.7$ wear seatbelts as the roller coaster negotiates its $6.7-\mathrm{m}$-radius loop at $9.5 \mathrm{~m} / \mathrm{s}$. At the top of the loop, what are the magnitude and direction of the force exerted on a $55-\mathrm{kg}$ rider (a) by the roller-coaster seat and (b) by the seatbelt?
(c) What would happen if the rider unbuckled at this point?

Key Concepts

Restraint Forces in Dynamic Systems

In dynamic systems such as roller coasters, restraint forces provided by structures like seats and seatbelts are crucial for ensuring that riders follow the intended path. These forces supplement the gravitational and normal forces to provide the proper net inward force required for circular motion. Understanding these forces is important, particularly when assessing the effects of removing one of the restraints, which can drastically alter the net force acting on the rider.

Free Body Diagram Analysis

A free body diagram is a tool used to visually represent all the forces acting on an object. In the context of circular motion, it is essential for identifying and resolving the individual forces (like gravitational force, normal force from the seat, and forces from restraints such as seatbelts) and understanding how they contribute to the net centripetal force necessary for the motion.

Newton’s Second Law in Circular Motion

Newton's Second Law, F = ma, is applied in circular motion by equating the net inward (centripetal) force to the product of mass and centripetal acceleration. This analysis helps in determining the sum of all the individual forces that must act together to provide the necessary centripetal force for maintaining the circular movement, taking into account forces such as gravity, normal force, and any additional restraint forces.

Centripetal Acceleration

Centripetal acceleration is the acceleration that is required to keep an object moving in a circle at a constant speed. It is always directed towards the center of the circular path and is computed using the formula a = v²/r, where v is the speed of the object and r is the radius of the circle. This concept is key in any problem involving circular motion since the force that maintains the object’s path is directly related to this acceleration.

Transcript

00:01 Okay, in problem 5 .45, it is an extension of, or a variation of example 5 .7, where a rider on a roller coaster, the great american revolution, goes around a loop -de -loop, and at the top of the coaster, it has a speed, the roller coaster car has a speed of 9 .5 meters per second.

00:32 The radius of the roller coaster at this of this loop -de -loop is 6 .7 meters.

00:41 So we need to know what the force by the roller coaster seat and the seat belt are and what would happen if the seat, if the rider were unbuckled.

00:51 So let's look at the forces first.

00:53 We know that if the roller coaster is upside down, the normal force, if there is one, is going to be pointing perpendicularly away from the surface.

01:02 There's a weight for sure.

01:05 So there's the force of gravity.

01:07 And if there's a seatbelt force, it's holding the rider in.

01:10 So it would be in the opposite direction.

01:12 So we sort of have to decide if there's a normal force or a seatbelt force because there won't be both because they oppose each other.

01:21 So what we can do to answer that question is look at newton's second law, essentially.

01:28 F -net is equal to ma, but when things are moving in a circle, the excessive.

01:32 Acceleration is v squared divided by r.

01:35 So if we can find the value of the net force, we can compare it to the force that we know is there, which is gravity, and then decide if we need a seatbelt force to balance it out to equal to mv squared over r or if we need a normal force.

01:49 So plugging in the values that they give us, the net force is equal to the mass, which is 55 kilograms, times the velocity, which was 9 .5 meters per second.

02:02 And that quantity is squared divided by the radius of the circle, which is 6 .7 meters.

02:13 So plugging those numbers in, we get a net force of 740 newtons.

02:21 So in order to go in a circle, a net force of 740 newtons is required...

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