As you know, banking a highway curve can help keep cars from skidding. Suppose that the radius of curvature of a given curve is r = 60 m, and that the recommended speed is v = 40 km/h . At what angle 0 should the curve be banked? Draw the free body diagram Forces Components Apply 2nd Newton’s Law Please follow required steps to better understand
The Correct Answer and Explanation is :
To calculate the angle at which the curve should be banked, we can apply the principles of circular motion and Newton’s second law.
Step 1: Understanding the forces
When a car is going around a banked curve, two forces act on it:
- The gravitational force (mg) acting vertically downward.
- The normal force (N) from the surface of the road, which is perpendicular to the road surface.
In addition to these, there is a centripetal force required for the car to travel in a circular path. This force is provided by the horizontal component of the normal force.
Step 2: Resolving the forces
- Vertical direction: The vertical forces must balance out since the car does not move vertically. Therefore, we have:
[
N \cos(\theta) = mg
] - Horizontal direction: The horizontal component of the normal force provides the centripetal force necessary for circular motion. This gives us:
[
N \sin(\theta) = \frac{mv^2}{r}
]
where ( v ) is the speed of the car, ( r ) is the radius of the curve, and ( m ) is the mass of the car.
Step 3: Relating the forces
We can eliminate the normal force ( N ) by dividing the second equation by the first equation:
[
\tan(\theta) = \frac{v^2}{rg}
]
Now, substitute the given values:
- ( v = 40 \, \text{km/h} = 11.11 \, \text{m/s} ) (convert km/h to m/s),
- ( r = 60 \, \text{m} ),
- ( g = 9.8 \, \text{m/s}^2 ).
Step 4: Solve for ( \theta )
Substitute the values into the equation:
[
\tan(\theta) = \frac{(11.11)^2}{60 \times 9.8}
]
[
\tan(\theta) = \frac{123.43}{588}
]
[
\tan(\theta) = 0.21
]
Now, calculate ( \theta ):
[
\theta = \tan^{-1}(0.21) \approx 11.9^\circ
]
Conclusion:
The curve should be banked at an angle of approximately 11.9° to prevent skidding at the recommended speed.
Let me generate the free body diagram to help visualize this.
Here is the free body diagram of the car on the banked curve. It shows the forces acting on the car and their components, with the angle of the banked curve approximately 11.9° as calculated earlier.
This diagram should help you better visualize how the forces are balanced and how the car’s motion is controlled around the curve. Let me know if you need further clarification!
