A special electronic sensor is embedded in the seat of a car that takes riders around a circular loop-the-loop ride at an amusement park. The sensor measures the magnitude of the normal force that the seat exerts on a rider. The loop-the-loop ride is in the vertical plane and its radius is 25 m. Sitting on the seat before the ride starts, a rider is level and stationary, and the electronic sensor reads 780 N. At the top of the loop, the rider is upside down and moving, and the sensor reads 380 N. What is the speed of the rider at the top of the loop?

Question

A special electronic sensor is embedded in the seat of a car that takes riders around a circular loop-the-loop ride at an amusement park. The sensor measures the magnitude of the normal force that the seat exerts on a rider. The loop-the-loop ride is in the vertical plane and its radius is 25 m. Sitting on the seat before the ride starts, a rider is level and stationary, and the electronic sensor reads 780 N. At the top of the loop, the rider is upside down and moving, and the sensor reads 380 N. What is the speed of the rider at the top of the loop?

Answer 1

To find the speed of the rider at the top of the loop, we can make use of the concept of centripetal force.

First, let's analyze the forces acting on the rider at the top of the loop. Since the rider is upside down, the only forces acting on them are gravity (mg) and the normal force (N).

At the top of the loop, the rider is experiencing two forces: gravitational force and the centripetal force. The gravitational force is pointing downwards and the normal force is pointing upwards. The centripetal force is the net force required to keep the rider moving in a circular path.

To begin, we need to determine the gravitational force acting on the rider. We know that the normal force at the top of the loop is 380 N. At this point, the normal force is equal to the centripetal force. Therefore, we can equate the normal force to the gravitational force at the top.

N = mg

380 N = mg

Next, we need to find the gravitational force acting on the rider. Rearranging the equation, we have:

mg = 380 N

Now, we can determine the speed of the rider at the top of the loop. The centripetal force can be calculated using the formula:

Centripetal Force (Fc) = (mass of the rider) x (velocity squared) / (radius of the loop)

Since the mass of the rider is not given, we can cancel it out by dividing both sides of the equation by m:

Fc/m = v²/r

Now, we can substitute the known values into the equation. The radius of the loop is given as 25 m, and we found that the gravitational force (mg) is 380 N.

380 N = v²/25 m

To find the speed, we can now solve for v. Multiplying both sides of the equation by 25 m:

380 N x 25 m = v²

9500 N·m = v²

Finally, taking the square root of both sides of the equation gives us the speed of the rider at the top of the loop:

v = √(9500 N·m)

Calculating the square root of 9500 N·m will yield the speed of the rider at the top of the loop.