Variably Solving for Minimum Tangential Speed

In summary, the problem is to find the minimum tangential speed needed for a rider on a carnival ride to not fall off when the bottom drops. The relevant equations are F=ma, F=un, and Ca=v^2/r. The attempted solution involves using V=sqrt (Ca*r), but it may be missing some variables. Additional information needed includes the normal reaction of the surface, the direction of the frictional force, and the necessary condition for the rider to not fall off.
  • #1
CandyApples
28
0

Homework Statement


A ride at a carnival rotates about a vertical axis. When spinning fast enough the bottom is dropped and the rider sticks to the side. Find the minimum tangential speed so that the rider does not fall. R=radius of circle, u= coefficient of static friction and g represents gravity. The answer must be in these terms.


Homework Equations


F = ma
F = un
Ca = v^2/r

The Attempt at a Solution


V = sqrt (Ca*r)

I think I am missing something here, as it is not with respect to the right variables. Any hints on what to do next, or if I am heading down the wrong path?
 
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  • #2
What is the normal reaction of the surface?
In which direction the frictional force acts?
What should the condition so that rider does not fall?
 
  • #3


Your attempt at a solution is on the right track. To find the minimum tangential speed, we need to consider the forces acting on the rider at the bottom of the ride. These forces include the normal force, the weight of the rider, and the frictional force. At the minimum tangential speed, the frictional force is equal to the maximum static friction force, which is given by uN, where u is the coefficient of static friction and N is the normal force. We can set this equal to the centripetal force, which is given by mv^2/r, where m is the mass of the rider, v is the tangential speed, and r is the radius of the circle.

So we have:

uN = mv^2/r

We can also express the normal force in terms of the weight of the rider, which is given by mg, where g is the acceleration due to gravity. So we have:

u(mg) = mv^2/r

Solving for v, we get:

v = sqrt(ugr)

This is the minimum tangential speed required for the rider to stick to the side of the ride. We can also express this in terms of the coefficient of static friction and the acceleration due to gravity:

v = sqrt(ugr)

So for example, if the radius of the ride is 10 meters, the coefficient of static friction is 0.5, and the acceleration due to gravity is 9.8 m/s^2, then the minimum tangential speed required would be:

v = sqrt(0.5 * 10 * 9.8) = 7.0 m/s

I hope this helps!
 

1. What is Variably Solving for Minimum Tangential Speed?

Variably Solving for Minimum Tangential Speed is a mathematical approach used to determine the minimum speed required for an object to travel along a curved path without slipping or losing contact with the surface.

2. Why is Variably Solving for Minimum Tangential Speed important?

This technique is important in many fields of science and engineering, including physics, mechanics, and robotics, as it helps determine the necessary speed for an object to safely and efficiently move along a curved path.

3. How is Variably Solving for Minimum Tangential Speed calculated?

The calculation involves using the equations of motion and the radius of curvature of the path to determine the minimum tangential speed. This speed is often denoted as vmin and is given by the equation vmin = √(g*r), where g is the gravitational acceleration and r is the radius of curvature.

4. What are some real-world applications of Variably Solving for Minimum Tangential Speed?

Variably Solving for Minimum Tangential Speed is commonly used in roller coasters, where it helps determine the minimum speed required for a train to safely travel around a loop without falling off the track. It is also used in automotive design to determine the minimum speed required for a car to safely navigate a turn without skidding.

5. What are the limitations of Variably Solving for Minimum Tangential Speed?

This approach assumes that the object is moving along a frictionless surface, which may not always be the case in real-world scenarios. It also does not take into account other factors such as air resistance, which can affect the actual minimum speed required for an object to move along a curved path.

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