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 17 m. Sitting on the seat before the ride starts, a rider is level and stationary, and the electronic sensor reads 820 N. At the top of the loop, the rider is upside down and moving, and the sensor reads 370 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 17 m. Sitting on the seat before the ride starts, a rider is level and stationary, and the electronic sensor reads 820 N. At the top of the loop, the rider is upside down and moving, and the sensor reads 370 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 use the principles of circular motion and the relationship between the forces involved.

First, let's consider the forces acting on the rider at the top of the loop. These forces include the gravitational force (mg) and the normal force (N) exerted by the seat on the rider.

When the rider is at the top of the loop, the net force acting on the rider must point toward the center of the circular path to keep the rider moving in a circular motion. This means that the net force is the difference between the gravitational force and the normal force:

Net force = mg - N

Since the rider is moving in a circular motion, we can relate the net force to the centripetal force (Fc), which is given by:

Fc = (mass of the rider) × (velocity of the rider at the top of the loop)^2 / (radius of the loop)

Substituting the value of net force into the equation for the centripetal force, we get:

mg - N = (mass of the rider) × (velocity of the rider at the top of the loop)^2 / (radius of the loop)

Now, let's substitute the given values into the equation:

mass of the rider = m (unknown)
velocity of the rider at the top of the loop = v (unknown)
radius of the loop = 17 m
gravity = 9.8 m/s^2

820 N - 370 N = m × v^2 / 17 m

450 N = m × v^2 / 17 m

To solve for v^2, we can multiply both sides of the equation by 17 m:

450 N × 17 m = m × v^2

7650 N·m = m × v^2

Next, we can cancel out the mass (m) on both sides of the equation:

7650 N·m = v^2

Finally, let's take the square root of both sides to find the velocity (v):

v = sqrt(7650 N·m)

Using a calculator, the velocity will be approximately:

v ≈ 87.58 m/s

Therefore, the speed of the rider at the top of the loop is approximately 87.58 m/s.