Centripetal Force at an Angle

In summary, the problem asks for the minimum radius of a rounded segment on a steep hill, so that cars traveling at 90 km/h will not leave the road. The solution involves setting up an equation where the centripetal force equals the force of gravity and finding the strictest requirement for the largest angle in the problem. The final answer is that the minimum radius must be greater than 68.8m.
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[SOLVED] Centripetal Force at an Angle

Homework Statement



The design of a new road includes a straight stretch that is horizontal and flat but that suddenly dips down a steep hill at 22 degrees. The transition should be rounded with what minimum radius so that cars traveling 90 km/h will not leave the road?

Homework Equations



v=(2*pi*R)/T
F=ma
a=(v^2)/R

The Attempt at a Solution



90 km/hr=25 m/s. I tried setting up an equation where the centripetal force equaled the force of gravity:(25^2)/r=9.8 sin 22 , but that gives me a radius of approximately 170.25m, when the correct answer is 63.8m.

EDIT: Solved. (25^2)/r=9.8
 
Last edited:
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  • #2
To flesh out OP’s solution:

In the rounded segment the car will travel along a vertical circle. In order to do so, a centripetal accceleration of ##mv^2/r## is required. In the limiting case where the car just avoids lifting off, the only force supplying acceleration in the radial direction is the radial component of the gravitational force, which equals ##mg\cos\theta##. It is therefore required that ##v^2/r < g\cos\theta## or, equivalently, ## r > v^2/g\cos\theta##. This requirement is strictest for the largest value of ##\theta## in the problem, ie, ##\theta = 22^\circ##. This gives ##r > 68.8## m.*

This assumes traveling at 25 m/s throughout the rounded segment. Assuming acceleration through the tangential component of gravity makes this somewhat worse, requiring larger radius of curvature. It is unclear why the problem author has only considered the requirement at the top of the segment.
 

Related to Centripetal Force at an Angle

1. What is centripetal force at an angle?

Centripetal force at an angle is the force that acts towards the center of a circular path, allowing an object to maintain a constant speed and direction while moving at an angle.

2. How is centripetal force at an angle calculated?

The formula for calculating centripetal force at an angle is Fc = mv^2/r, where Fc is the centripetal force, m is the mass of the object, v is the velocity, and r is the radius of the circular path.

3. What factors affect the magnitude of centripetal force at an angle?

The magnitude of centripetal force at an angle is affected by the mass of the object, the speed at which it is moving, and the radius of the circular path. The greater the mass, speed, or radius, the greater the centripetal force required.

4. Can centripetal force at an angle be greater than the weight of the object?

Yes, it is possible for the magnitude of centripetal force at an angle to be greater than the weight of the object. This occurs when the object is moving at a high enough speed or on a small enough radius to require a large centripetal force.

5. How does centripetal force at an angle relate to Newton's laws of motion?

Centripetal force at an angle is an example of Newton's first law of motion, which states that an object will continue to move in a straight line at a constant speed unless acted upon by an external force. In this case, the centripetal force is the external force that causes the object to move in a circular path.

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