Minimum speed at the circular track

In summary, the minimum speed required at a circular track depends on factors such as the radius of the track, angle of bank, and coefficient of friction. The larger the radius, the higher the minimum speed needed, while a higher angle of bank and coefficient of friction can lower the minimum speed required. Other methods such as using centripetal force, Newton's second law, and energy conservation can also be used to calculate the minimum speed, but the formula v = √(rgtanθ) is the most commonly used and accurate for most scenarios.
  • #1
inky
99
1

Homework Statement



What is the minimum speed of the car must have at the top of the loop? There are three methods. I would like to know which method or answer is correct.

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Homework Equations



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The Attempt at a Solution



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  • #2
I would choose the third one,provided that H> 2r.
 
  • #3
rock.freak667 said:
I would choose the third one,provided that H> 2r.

Thanks a lot. Why didn't consider the force at the top?
 
  • #4
I think if H> 2r, then the speed will be enough to provide the centripetal force.
 
  • #5


I would first like to clarify that the minimum speed at the top of the loop is not a fixed value, but rather a range of values depending on the specific parameters of the track and car. Therefore, it is important to clearly define the variables and assumptions used in the calculations.

Method 1: Conservation of Energy
In this method, the minimum speed at the top of the loop can be calculated using conservation of energy principles. The minimum speed would be equal to the square root of (2gr), where g is the acceleration due to gravity and r is the radius of the loop. However, this method assumes that there is no friction or air resistance, which may not be the case in a real-world scenario.

Method 2: Centripetal Force
Using centripetal force equations, the minimum speed can be calculated by equating the centripetal force to the weight of the car at the top of the loop. This method takes into account the effects of friction and air resistance, but may not be accurate if the track has uneven surfaces or the car is not moving in a perfectly circular path.

Method 3: Maximum Height
The minimum speed can also be calculated by considering the maximum height of the loop. Using basic kinematic equations, the minimum speed would be equal to the square root of (2gR), where R is the maximum height of the loop. This method assumes that the car is not moving too fast to maintain contact with the track at the top of the loop.

In conclusion, all three methods may provide different values for the minimum speed at the top of the loop and the correct answer would depend on the specific conditions of the track and car. It is important to carefully consider all variables and assumptions before determining the minimum speed.
 

Related to Minimum speed at the circular track

1. What is the minimum speed required at a circular track?

The minimum speed required at a circular track depends on various factors such as the radius of the track, the angle of bank, and the coefficient of friction. However, in general, the minimum speed required to maintain circular motion at a track without slipping is given by the equation v = √(rgtanθ), where v is the minimum speed, r is the radius of the track, g is the acceleration due to gravity, and θ is the angle of bank.

2. How does the radius of the track affect the minimum speed?

The radius of the track directly affects the minimum speed required to maintain circular motion. A larger radius will require a higher minimum speed, while a smaller radius will require a lower minimum speed. This is because the centripetal force required to keep an object moving in a circular path increases with the radius of the track.

3. What is the role of the angle of bank in determining the minimum speed?

The angle of bank, also known as the banking angle, is the angle at which the track is tilted towards the center of the circle. This angle plays a crucial role in determining the minimum speed required on a circular track. A higher angle of bank will require a lower minimum speed, while a lower angle of bank will require a higher minimum speed. This is because a higher angle of bank helps to provide a greater component of the normal force, which is essential for maintaining circular motion without slipping.

4. How does the coefficient of friction affect the minimum speed at a circular track?

The coefficient of friction is a measure of the frictional force between two surfaces in contact. In the case of a circular track, it affects the maximum speed at which an object can travel without slipping. A higher coefficient of friction will allow for a higher minimum speed, while a lower coefficient of friction will require a lower minimum speed. This is because a higher coefficient of friction provides a larger centripetal force, which is necessary to maintain circular motion.

5. Can the minimum speed at a circular track be calculated using other methods?

Yes, there are various methods to calculate the minimum speed at a circular track. Apart from the equation v = √(rgtanθ), other methods such as using the concepts of centripetal force, Newton's second law, and energy conservation can also be used to calculate the minimum speed. However, the formula mentioned above is the most commonly used and provides accurate results for most scenarios.

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