banked curves and circular motion explained

PhysicsHigh

28 Oct 202205:35

EducationalLearning

32 Likes 10 Comments

TLDRIn this educational video, Paul from Physics High succinctly explains the physics behind banked curves, a concept where roads are angled to allow vehicles to travel faster around a curve. He demonstrates how the centripetal force, resulting from the car's weight (mg) and the normal force, keeps the car in circular motion. By simplifying the scenario without friction, Paul uses vector analysis and component analysis to derive the formula for the velocity (V) of an object in circular motion, which is independent of mass and depends on the radius (r) and angle (Theta) of the banked curve. The video also touches on the complexity added by friction and encourages viewers to watch a more detailed video for further understanding.

Takeaways

🚗 The video discusses forces involved with banked curves, which are curved tracks designed to allow vehicles to travel faster around turns.
🎥 Paul, from Physics High, is the presenter who provides an overview of banked curves and their physics.
🔗 A more detailed video on banked curves is available, which this short version complements.
📐 Banked curves are circular tracks with the road surface at an angle to increase speed around the curve.
🧲 The primary forces considered in the analysis are gravity (mg) and the normal force, with friction being ignored for simplicity.
📚 The relationship between the forces is visualized through vector diagrams and component analysis.
📉 The centripetal force, which keeps the car in circular motion, is the resultant of the normal force and gravitational force.
🔍 The tangent of the angle theta (θ) is equal to the ratio of centripetal force (Mv^2/r) to the gravitational force (mg).
📘 The formula derived for velocity (V) is V = √(g * r * tan(θ)), showing that it is independent of the car's mass.
🔄 If the velocity increases, the car will move up the bank; if it decreases, it will move down, indicating the role of velocity in maintaining circular motion.
🚫 Adding friction to the analysis complicates the scenario, which is typically not required in high school exams but is covered in more detail in the full video.

Q & A

What is a banked curve in the context of physics?
-A banked curve is a circular track designed for vehicles to travel around at higher speeds. The road surface is angled to increase the opportunity for faster travel around the curve.
What is the purpose of banking a curve on a track?
-Banking a curve on a track is done to increase the speed at which a vehicle can safely navigate the curve, by providing additional centripetal force through the angle of the bank.
What are the two basic forces considered in the analysis of banked curves without friction?
-The two basic forces considered are the force due to the weight of the car (mg) and the normal force acting perpendicular to the road surface.
How does the centripetal force in a banked curve scenario relate to the forces of weight and normal force?
-The centripetal force in a banked curve scenario is the result of the weight force and the normal force acting together. It is the net force that keeps the car moving in a circular path.
What mathematical relationship can be derived from the forces acting on a car in a banked curve?
-The mathematical relationship derived is that the tangent of the angle of the bank (theta) is equal to the centripetal force (which is MV^2/r) divided by the weight force (mg). Simplifying this gives V^2 = g*r*tan(theta), where V is the velocity of the car.
Why is the mass of the car not included in the formula for the velocity required to stay in circular motion on a banked curve?
-The mass of the car is not included in the formula because the velocity required for an object to stay in circular motion on a banked curve is independent of its mass, given that friction is ignored.
What happens to the car if the velocity increases or decreases on a banked curve?
-If the velocity increases, the car will slide up the hill due to the increased normal force. If the velocity decreases, the car will slide down the hill due to the reduced normal force.
What is vector analysis and how is it used in the context of banked curves?
-Vector analysis is a method of breaking down forces into their component parts. In the context of banked curves, it is used to determine the relationship between the normal force, weight, and centripetal force acting on a car.
What is component analysis and how does it relate to the normal force in a banked curve scenario?
-Component analysis is a technique used to resolve vectors into their orthogonal components. In a banked curve scenario, it is used to separate the normal force into horizontal and vertical components, where the horizontal component provides the centripetal force.
What additional complexity is introduced when friction is considered in the analysis of banked curves?
-When friction is considered, the analysis becomes more complex as it introduces an additional vector force that must be added to the component analysis. The frictional force is equal to the coefficient of friction (mu) times the normal force and can act either up or down the plane.
What is the significance of the angle theta in the context of banked curves?
-Theta is the angle of the banked curve. It is significant because it determines the relationship between the normal force, weight, and the required centripetal force for a car to maintain circular motion without sliding.

Outlines

00:00

🚗 Introduction to Banked Curves and Forces

Paul from Physics High introduces the concept of banked curves, explaining the forces involved when a car navigates a circular track. He simplifies the scenario by ignoring frictional forces to focus on the primary forces: the car's weight (mg) and the normal force from the road. He illustrates how these forces combine to create the centripetal force necessary for circular motion. Paul uses both vector analysis and component analysis to derive the formula for the velocity of an object in circular motion on a banked curve, which is independent of the object's mass. The formula is V = √(g * r * tan(θ)), where g is the acceleration due to gravity, r is the radius of the curve, and θ is the angle of the banked curve.

05:02

🔍 Adding Friction to Banked Curve Analysis

In the second paragraph, Paul touches on the complexity added when friction is considered in the analysis of banked curves. He mentions that frictional force can act in two different scenarios, either up or down the plane, leading to two distinct cases to explore. The frictional force is described as being equal to the coefficient of friction (μ) times the normal force. Although not detailed in this short version of the video, Paul refers viewers to a more comprehensive video where he explains how to include friction in the component analysis of banked curves. He concludes by hoping the explanation aids in understanding banked curves and signs off with a reminder of his affiliation with Physics High.

Mindmap

Keywords

💡Banked curves

Banked curves refer to the design of a circular road or track where the outer side is elevated relative to the inner side. This design is used to increase the speed at which vehicles can safely navigate a turn by utilizing the incline to provide additional centripetal force. In the video, Paul discusses how banked curves allow vehicles to travel faster around a curve by increasing the centripetal force acting on the vehicle.

💡Centripetal force

Centripetal force is the inward force that keeps an object moving in a circular path. It is essential for maintaining circular motion and is provided by the net force acting towards the center of the circle. In the context of the video, the centripetal force is the result of the combined effect of the normal force and the gravitational force acting on the car as it travels around the banked curve.

💡Normal force

The normal force is the perpendicular force exerted by a surface that supports the weight of an object resting on it. In the video, the normal force acts on the car in the direction perpendicular to the road surface. It is one of the two primary forces considered when analyzing the physics of a banked curve, along with the gravitational force.

💡Gravitational force

Gravitational force, often represented as 'mg' where 'm' is the mass of the object and 'g' is the acceleration due to gravity, is the force that attracts a body towards the center of the Earth. In the script, the gravitational force acts vertically downwards and is a key component in the analysis of forces acting on a car in a banked curve.

💡Frictional forces

Frictional forces are resistive forces that oppose the relative motion of two surfaces in contact. In the video, Paul initially ignores frictional forces to simplify the analysis of banked curves. However, he also mentions that including friction would add complexity to the analysis, as it would require considering the frictional force vector acting either up or down the plane.

💡Vector analysis

Vector analysis is a method used to break down and resolve complex forces into their component parts. In the script, Paul uses vector analysis to resolve the gravitational force and the normal force into components that can be added to find the net centripetal force acting on the car. This method helps in understanding the relationship between the forces and the motion of the car around the banked curve.

💡Component analysis

Component analysis is similar to vector analysis but focuses on breaking down forces into their horizontal and vertical components. In the video, Paul uses component analysis to further explain how the normal force has both a horizontal component (which contributes to the centripetal force) and a vertical component (which balances the gravitational force).

💡Tangential

Tangential in this context refers to something that is related to or touches a circle at a single point, such as the direction of the net force or velocity that is tangent to the curve. Paul uses the term 'tangential' when discussing the relationship between the angle of the banked curve and the resultant centripetal force, which is tangent to the circular path of the car.

💡Velocity

Velocity is a vector quantity that refers to the speed of an object in a particular direction. In the video, Paul explains that the velocity of an object in circular motion on a banked curve is independent of its mass. The formula derived in the script, V = sqrt(gr tan(Theta)), shows that the velocity depends on the radius of the curve, the acceleration due to gravity, and the angle of the banked curve.

💡Mass

Mass is a measure of the amount of matter in an object, typically measured in kilograms. In the context of the video, Paul points out that the mass of the car does not affect its velocity when traveling in circular motion on a banked curve. This is an important observation as it simplifies the physics involved and shows that the velocity is determined by other factors, such as the curve's geometry.

Highlights

Introduction to banked curves and their purpose in allowing vehicles to travel faster around curves.

Explanation of centripetal force resulting in a car's circular motion on a banked curve.

Simplification of the problem by ignoring frictional forces for the initial analysis.

Identification of two basic forces acting on a car: weight (mg) and normal force.

Vector diagram used to combine the forces acting on the car.

Mathematical relationship derived from the forces: tan(theta) = centripetal force / mg.

Velocity formula for an object in circular motion: V = sqrt((g * r * tan(theta)) / r).

Independence of mass in the formula for velocity in banked curves.

Behavior of the car with changes in velocity: sliding up or down the hill.

Introduction to component analysis of forces.

Breaking down the normal force into vertical and horizontal components.

Derivation of the centripetal force from the horizontal component of the normal force.

Vertical component of the normal force balancing the weight of the car.

Component analysis leading to the same velocity formula as vector analysis.

Complexity introduced by adding friction to the analysis.

Mention of frictional force direction and its impact on the analysis.

Brief mention of a more detailed video on banked curves with friction.

Conclusion and sign-off by Paul from Physics High.

Transcripts

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Takeaways

Q & A

What is a banked curve in the context of physics?

What is the purpose of banking a curve on a track?

What are the two basic forces considered in the analysis of banked curves without friction?

How does the centripetal force in a banked curve scenario relate to the forces of weight and normal force?

What mathematical relationship can be derived from the forces acting on a car in a banked curve?

Why is the mass of the car not included in the formula for the velocity required to stay in circular motion on a banked curve?

What happens to the car if the velocity increases or decreases on a banked curve?

What is vector analysis and how is it used in the context of banked curves?

What is component analysis and how does it relate to the normal force in a banked curve scenario?

What additional complexity is introduced when friction is considered in the analysis of banked curves?

What is the significance of the angle theta in the context of banked curves?