The weight of passengers on a roller coaster increases by 50% as the car goes through a dip with a 30 m radius of curvature. What is the car's speed at the bottom of the dip?

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Hi everyone in this track. This problem, we are going to determine the magnitude of the bus's velocity at the bottom of the track. We will have a bus loaded with students going through a circular track within a radius of 19 m. And when the bus reached its lowest point, a student actually claimed that her weight had doubled. So we wanna find the magnitude of the bus's velocity at the bottom of the track that will actually cause this phenomenon to happen. So first, what we wanna do is to draw the system of our problem. So first, we have the circular track here and I'm going to represent the buzz or essentially the student with the sphere right here. In this case, the student or the boss will experience two different forces. The first one is going to be the normal force which will be perpendicular to the track or the circular track, which is going to be uh pointing upwards. So that will be represented by the red arrow. And the end second is going to be the weight, which will be the weight of the student actually. And that will be represented by this blue arrow pointing downwards and that will be M multiplied by G I am going to define that our vertical direction axis or the Y axis, the y vertical direction has to be positive in the upward direction. All right. So according to the problem statement, when the bus reached its lowest point, a student claimed that her weight had doubled. So in this case, we're going to focus on the student itself. And what the student felt in this phenomenon is actually the normal force N. So the normal force N on her apparent weight from this information is actually going to be double or essentially two times the gravitational force or two times the actual weight. So two times M G and I'm going to refer to this to our first equation. All right. So the way we want to solve for this problem is to actually by applying Newton's second law. So according to Newton second law, the summation or the total of all the forces acting on the radial direction on a particular system, which in this case, is going to be the student will, it goes to the mass of the student or the mass of our system multiplied by A R. In this case, we have both the weight and also the normal force, but the weight is pointing downward. So we wanna uh put a negative sign in front of it. So negative M G plus the normal force will equals to M multiplied by A R just like. So we want to actually uh recall that A R is going to equals to the linear velocity divided by the R or the radius of the trajectory itself. So in this case, this philosophy right here, the linear velocity is what we are interested in finding, we wanna substitute this into our Newton second law equation so that it's going to be negative M G plus N equals to M multiplied by V squared divided by R. Next. You want to substitute equation one into our Newton Second law expression that will give us a negative M G plus two M G equals M multiplied by V squared divided by R. Uh That will actually equals to M G equals to M multiplied by V squared divided by R. And we can cross out both MS in both sides. And that will give us a value or an expression for our velocity which will equals to the square root of G multiplied by R just like. So, so we can actually start substituting all the values that we know into our expression. So the velocity is then going to equals to the gravitational acceleration which is 9.81 m per second squared multiplied that by the radius which is going to be m. And that will give us a velocity value of 13.7 m/s. So the velocity at the bottom of the track of the student, which will equals to the magnitude of the philosophy of the bus at the bottom of the track will equals to 13.7 m per second, which will correspond to option B in our answer choices. So answer B is going to be the answer to our problem statement with the philosophy of the bus at the bottom of the track being 13.7 m per second. So that'll be all for this particular video. If you guys still have any sort of confusion, please make sure to check out our other lesson videos on similar topic and that will be it. Thank you.

The weight of passengers on a roller coaster increases by 50% as the car goes through a dip with a 30 m radius of curvature. What is the car's speed at the bottom of the dip?

Verified Solution

Key Concepts

Centripetal Force

Weight and Normal Force

Kinematics and Circular Motion