A curve of radius 72 m is banked for a design speed of 120 km/h. If the coefficient of static friction is 0.31 (wet pavement), at what range of speeds can a car safely handle the curve?
Express your answers using two significant figures

To find the range of speeds at which a car can safely navigate a banked curve with static friction, we can use the physics of circular motion and the following key variables:

- \( r = 72 \, \text{m} \) (radius of the curve),
- \( v_{\text{design}} = 120 \, \text{km/h} = 33.33 \, \text{m/s} \) (design speed),
- \( \mu_s = 0.31 \) (coefficient of static friction),
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity).

First, we calculate the banking angle \( \theta \) from the design speed using the formula for a banked curve without friction:
\[
\tan \theta = \frac{v_{\text{design}}^2}{r g}
\]

Now, for the range of speeds, we need to account for the effects of static friction. We can derive the two limiting cases for speed:

1. **Maximum Speed (with friction assisting the car)**
   \[
   v_{\text{max}}^2 = \frac{r g (\tan \theta + \mu_s)}{1 - \mu_s \tan \theta}
   \]

2. **Minimum Speed (with friction opposing the car)**
   \[
   v_{\text{min}}^2 = \frac{r g (\tan \theta - \mu_s)}{1 + \mu_s \tan \theta}
   \]

Let's calculate these values step by step.The car can safely handle the curve at speeds between **88 km/h** and **180 km/h** (rounded to two significant figures).

Would you like further details on the calculations or have any other questions?

Here are five related questions to deepen your understanding:
1. What is the role of friction in determining maximum and minimum speeds?
2. How would the safe speed range change if the curve radius were different?
3. What happens if the pavement is dry with a higher coefficient of static friction?
4. Why is the design speed important for banked curves without friction?
5. How does the banking angle affect the centripetal force?

**Tip:** The greater the banking angle, the less reliance there is on friction for maintaining safe speeds on a curve.

A curve of radius 72 m is banked for a design speed of 120 km/h. If the coefficient of static friction is 0.31 (wet pavement), at what range of speeds can a car safely handle the curve? Express your answers using two significant figures.

Learn how to calculate the safe speed range for a car on a banked curve with a radius of 72 m, a design speed of 120 km/h, and a coefficient of static friction of 0.31 (wet pavement). This solution involves concepts of circular motion, centripetal force, and friction. Suitable for advanced high school or undergraduate physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation