# Problem 7.46: Riding a Loop-the-loop

• annabelx4
In summary, the problem involves a car in an amusement park ride starting from rest at a height above the bottom of a loop and moving around the loop without falling off at the top. The minimum value of is 4R and the speed of the passengers at the end of a horizontal diameter can be calculated using the equation mgh = 1/2 mv^2. When plugging in the given values, the speed is found to be 14.1 m/s.
annabelx4

## Homework Statement

A car in an amusement park ride rolls without friction around the track shown in the figure . It starts from rest at point at a height above the bottom of the loop. Treat the car as a particle.

What is the minimum value of (in terms of ) such that the car moves around the loop without falling off at the top (point )?

If the car starts at height 4.00 and the radius is = 25.0 , compute the speed of the passengers when the car is at point , which is at the end of a horizontal diameter.

mgh = 1/2 mv^2

## The Attempt at a Solution

we know that h = 2 R because it's the diameter but other than that i am lost

. For the minimum value, I believe it is 4R. For the speed, we use the equation mgh = 1/2 mv^2. mgh = 1/2 mv^2(4)(9.8)(25) = 1/2 (20)(v^2)1960 = 10v^2196 = v^214.1 m/s = v

The minimum value of h, in terms of R, can be found by equating the gravitational potential energy at the top of the loop (point C) to the kinetic energy at the bottom of the loop (point A). This can be represented by the equation mgh = 1/2 mv^2, where m is the mass of the car, g is the acceleration due to gravity, h is the height at the top of the loop, and v is the speed of the car at the bottom of the loop.

Solving for h, we get h = 1/2 v^2/g. Since the car starts at height 4.00 and the radius is 25.0, we can substitute these values to get 4.00 = 1/2 v^2/g. Solving for v, we get v = √(8gh), where g = 9.8 m/s^2.

To find the speed of the passengers at point B, which is at the end of a horizontal diameter, we can use the conservation of energy principle. At point B, the car is at the same height as point A, so the potential energy is the same. However, the kinetic energy at point B is now in the form of both translational and rotational energy. We can represent this as 1/2 mv^2 + 1/2 Iω^2, where I is the moment of inertia and ω is the angular velocity.

Since the car is treated as a particle, we can simplify this to 1/2 mv^2 + 1/2 mv^2 = mv^2. Plugging in the values, we get 4.00 = 1/2 mv^2. Solving for v, we get v = √(8gh), which is the same as the speed at the bottom of the loop.

In conclusion, the minimum value of h in terms of R is 1/2 v^2/g, and the speed of the passengers at point B is √(8gh).

## 1. How does the speed of the rider affect their ability to complete the loop?

The speed of the rider is a crucial factor in determining whether they can successfully complete the loop. If the rider's speed is too slow, they may not have enough centrifugal force to counteract the force of gravity and make it through the loop. On the other hand, if the rider's speed is too fast, they may experience excessive g-forces that can be dangerous.

## 2. What is the minimum height required for a loop-the-loop to be safe?

The minimum height required for a loop-the-loop to be safe depends on several factors such as the speed of the rider, the radius of the loop, and the strength of the materials used to construct the track. Generally, a height of at least 15 meters is recommended for a safe loop-the-loop.

## 3. Can a loop-the-loop be built without any support structures?

No, a loop-the-loop cannot be built without any support structures. The loop requires a strong support structure to withstand the forces exerted on it by the rider and the track itself. Without support structures, the loop would collapse under the weight and force of the rider.

## 4. What is the role of centripetal force in riding a loop-the-loop?

Centripetal force is the force that acts towards the center of the loop and keeps the rider in a circular motion. Without centripetal force, the rider would continue in a straight line and not be able to complete the loop. The rider's speed and the curvature of the loop determine the amount of centripetal force needed.

## 5. Are there any safety measures in place to prevent accidents while riding a loop-the-loop?

Yes, there are several safety measures in place to prevent accidents while riding a loop-the-loop. These include designing the loop with a minimum height and radius for safe completion, using strong and durable materials, ensuring the rider is properly restrained, and conducting regular safety inspections and maintenance on the track. In addition, riders must also follow safety guidelines and use proper protective gear to minimize the risk of accidents.

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