Car On Banked Track: Physics Problem Solved

Hey guys! Ever wondered how cars can zoom around a banked track without flying off? It's all thanks to some cool physics principles at play. Let's break down a classic problem involving a car on a banked track and see how it all works. This article will provide a comprehensive explanation of how to approach and solve this type of physics problem.

Understanding the Problem

So, we've got a car with mass m cruising around a circular track. This track isn't flat; it's banked at an angle of 37 degrees. The track has a radius of 50 meters, and the coefficient of static friction between the tires and the road is 0.5. Our mission is to figure out the car's maximum speed without it skidding off the track. Sounds like fun, right? This problem combines concepts from circular motion, forces, and friction, making it a great exercise for understanding how these principles interact. To solve this problem effectively, we need to consider all the forces acting on the car. These forces include gravity, the normal force exerted by the track, and the force of static friction. By carefully analyzing these forces and applying Newton's second law, we can determine the maximum speed the car can maintain without skidding.

Identifying the Forces

Let's pinpoint the forces acting on the car. First up, we have gravity, pulling the car straight down. Then there's the normal force, pushing the car perpendicular to the track's surface. And don't forget friction, which acts parallel to the track, preventing the car from sliding. The interplay of these forces is what keeps the car safely on its path. Visualizing these forces with a free-body diagram is super helpful. Draw the car as a point, and then draw arrows representing each force, making sure to indicate their directions and points of application. This visual aid will make it easier to resolve the forces into their components and apply Newton's laws correctly. In addition to these primary forces, it's important to understand how each force contributes to the car's motion. Gravity, for instance, has both a vertical and a horizontal component relative to the banked track. The normal force also has vertical and horizontal components, and these components play a crucial role in providing the centripetal force needed for circular motion. Friction, depending on its direction, can either assist in providing the necessary centripetal force or counteract it.

Breaking Down the Forces

Now, let's get mathematical. We need to break down the normal force and friction into their horizontal and vertical components. This will help us analyze how these forces contribute to the car's motion in each direction. We can use trigonometry (remember sine and cosine?) to find these components. For the normal force (${N}$), the vertical component is ${N \cos(37^\circ)}$, and the horizontal component is ${N \sin(37^\circ)}$. Similarly, for the friction force (${f}$), the vertical component is ${f \sin(37^\circ)}$, and the horizontal component is ${f \cos(37^\circ)}$. Breaking down the forces into components allows us to apply Newton's second law separately in the vertical and horizontal directions. This is a crucial step in solving the problem because it simplifies the analysis and allows us to relate the forces to the car's acceleration in each direction. In the vertical direction, the sum of the forces must be zero since the car is not accelerating vertically. In the horizontal direction, the sum of the forces must equal the centripetal force required to keep the car moving in a circle.

Applying Newton's Laws

Time for Newton's Second Law! In the vertical direction, the forces must balance out because the car isn't flying up or sinking down. So, we have: ${N \cos(37^\circ) + f \sin(37^\circ) = mg}$ In the horizontal direction, the forces provide the centripetal force needed for circular motion: ${N \sin(37^\circ) + f \cos(37^\circ) = \frac{mv^2}{r}}$

These two equations are the key to solving for the car's maximum speed. By carefully manipulating these equations, we can eliminate the normal force and friction force, allowing us to solve for the velocity ${v}$. This is a common technique in physics problem-solving, where we use multiple equations to eliminate unknowns and isolate the variable we want to find. To apply Newton's laws effectively, it's essential to understand the relationship between force, mass, and acceleration. Newton's second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the acceleration is the centripetal acceleration, which is directed towards the center of the circular path. The centripetal acceleration is given by ${a = \frac{v^2}{r}}$, where ${v}$ is the speed of the car and ${r}$ is the radius of the circular path.

Solving for Maximum Speed

We want to find the maximum speed, so we need to consider the maximum static friction force, which is given by ${f = \mu_s N}$. Plugging this into our equations, we get: ${N \cos(37^\circ) + \mu_s N \sin(37^\circ) = mg}$ ${N \sin(37^\circ) + \mu_s N \cos(37^\circ) = \frac{mv^2}{r}}$

Now we can solve for ${N}$ from the first equation and substitute it into the second equation to find ${v}$. After some algebraic gymnastics, we arrive at: ${v = \sqrt{gr \frac{\sin(37^\circ) + \mu_s \cos(37^\circ)}{\cos(37^\circ) - \mu_s \sin(37^\circ)}}}$

Plugging in the values (g = 9.8 m/s², r = 50 m, ${\mu_s}$ = 0.5, sin(37°) = 0.6, cos(37°) = 0.8), we get: ${v \approx 25.7 \text{ m/s}}$

So, the maximum speed the car can maintain without skidding is approximately 25.7 meters per second. This solution demonstrates how the combination of banked tracks and static friction allows cars to achieve higher speeds in circular motion without losing control. The algebraic manipulation involved in solving for the velocity requires careful attention to detail. It's important to double-check each step to ensure that no errors are made. The final result provides valuable insight into the physics of circular motion and the factors that influence the maximum speed achievable on a banked track.

Final Thoughts

And there you have it! By understanding the forces at play and applying Newton's laws, we can solve even the trickiest physics problems. Remember, drawing a free-body diagram and breaking down forces into components is crucial for success. Keep practicing, and you'll become a physics pro in no time! The principles we've discussed here have wide-ranging applications in engineering and design, from designing safer roads to optimizing the performance of race cars. Understanding these concepts allows us to create technologies that improve our lives and push the boundaries of what's possible.

Key takeaways:

• Always start with a clear free-body diagram.
• Break forces into components to simplify analysis.
• Apply Newton's laws in each direction.
• Don't forget the role of friction!

Keep exploring, keep questioning, and keep learning! This approach not only helps in solving physics problems but also fosters a deeper understanding of the world around us. Happy Physics-ing! Understanding physics concepts such as forces, friction, and motion not only helps in solving academic problems but also enables us to analyze and understand real-world phenomena. By applying these principles, we can make informed decisions and design solutions to improve safety and efficiency in various aspects of our lives.