Circular Motion - CBSE Notes for Class 11 Physics

H1 Circular Motion 12th Std Pdf Download --- --- H2 What is circular motion? H3 Definition and examples of circular motion H3 Types of circular motion: uniform and non-uniform H3 Parameters of circular motion: angular displacement, angular velocity, angular acceleration H2 How to analyze circular motion? H3 Centripetal force and acceleration H3 Newton's laws of motion and circular motion H3 Free-body diagrams and equations of motion for circular motion H2 How to solve problems on circular motion? H3 Steps to solve problems on circular motion H3 Common scenarios and applications of circular motion H4 Horizontal circular motion: car on a curved road, bucket of water, etc. H4 Vertical circular motion: roller coaster, loop-the-loop, etc. H4 Conical pendulum, banked road, etc. H2 How to download pdf files on circular motion 12th std? H3 Benefits of pdf files for learning circular motion 12th std H3 Sources and links to download pdf files on circular motion 12th std H4 Lecture notes and slides on circular motion 12th std H4 Textbooks and reference books on circular motion 12th std H4 Sample papers and solutions on circular motion 12th std Here is the article based on the outline: # Circular Motion 12th Std Pdf Download Circular motion is one of the most important topics in physics for students of class 12. It involves the study of the motion of objects along a curved path or a circle. In this article, we will learn about the concepts, formulas, and methods to analyze and solve problems on circular motion. We will also provide some sources and links to download pdf files on circular motion 12th std for your convenience. ## What is circular motion? Circular motion is a type of motion in which an object moves along a fixed distance from a fixed point, called the center of the circle. The object traces out a circle or a part of a circle as it moves. ### Definition and examples of circular motion Mathematically, we can define circular motion as the motion of an object whose position vector r always makes a constant angle with a fixed axis. Alternatively, we can define it as the motion of an object whose projection on a fixed plane always moves along a circle. Some examples of circular motion are: - The rotation of the earth around its axis - The revolution of the moon around the earth - The orbit of a satellite around a planet - The spinning of a wheel or a fan - The swinging of a pendulum ### Types of circular motion: uniform and non-uniform Circular motion can be classified into two types: uniform and non-uniform. Uniform circular motion is the circular motion in which the object moves with a constant speed along the circle. In this case, the object has a constant angular velocity (the rate of change of angular displacement) and zero angular acceleration (the rate of change of angular velocity). Non-uniform circular motion is the circular motion in which the object does not move with a constant speed along the circle. In this case, the object has a varying angular velocity and a non-zero angular acceleration. ### Parameters of circular motion: angular displacement, angular velocity, angular acceleration To describe the position, velocity, and acceleration of an object moving in a circle, we need to use some parameters that are different from those used for linear motion. These parameters are: - Angular displacement: It is the angle subtended by the position vector r at the center of the circle. It is measured in radians (2π radians = 360). It tells us how much the object has rotated from its initial position. - Angular velocity: It is the rate of change of angular displacement with respect to time. It is measured in radians per second (rad/s). It tells us how fast the object is rotating. - Angular acceleration: It is the rate of change of angular velocity with respect to time. It is measured in radians per second squared (rad/s^2). It tells us how fast the object is speeding up or slowing down its rotation. The relation between these parameters and their linear counterparts (displacement, velocity, and acceleration) is given by: - s = rθ, where s is the linear displacement, r is the radius of the circle, and θ is the angular displacement. - v = rω, where v is the linear velocity, r is the radius of the circle, and ω is the angular velocity. - a = rα, where a is the linear acceleration, r is the radius of the circle, and α is the angular acceleration. ## How to analyze circular motion? To analyze circular motion, we need to apply the concepts of force and acceleration to the object moving in a circle. We also need to use some diagrams and equations that help us to simplify and solve the problems. ### Centripetal force and acceleration One of the key concepts in circular motion is that of centripetal force and acceleration. Centripetal means "towards the center". Centripetal force is the net force acting on an object moving in a circle that makes it move towards the center of the circle. Centripetal acceleration is the acceleration of an object moving in a circle that makes it change its direction towards the center of the circle. The magnitude of centripetal force and acceleration is given by: - Fc = mv^2/r, where Fc is the centripetal force, m is the mass of the object, v is its linear speed, and r is the radius of the circle. - ac = v^2/r, where ac is the centripetal acceleration, v is the linear speed, and r is the radius of the circle. Alternatively, we can use angular velocity instead of linear speed and write: - Fc = mω^2r, where Fc is the centripetal force, m is the mass of the object, ω is its angular velocity, and r is the radius of the circle. - ac = ω^2r, where ac is the centripetal acceleration, ω is the angular velocity, and r is the radius of the circle. The direction of centripetal force and acceleration is always perpendicular to the velocity vector and towards the center of the circle. Therefore, they do not change the magnitude of velocity but only its direction. ### Newton's laws of motion and circular motion To find out what forces are responsible for providing centripetal force and acceleration to an object moving in a circle, we need to apply Newton's laws of motion. According to Newton's second law of motion, - Fnet = ma, where Fnet is the net force acting on an object, m is its mass, and a is its acceleration. For an object moving in a circle with constant speed (uniform circular motion), we can write: - Fnet = Fc = mac = mv^2/r This means that for an object to move in a circle with constant speed, there must be a net force acting on it that equals its centripetal force. This net force can be provided by various sources such as gravity, tension, friction, normal force, etc. depending on the situation. For an object moving in a circle with varying speed (non-uniform circular motion), we can write: - Fnet = mac + mat This means that for an object to move in a circle with varying speed, there must be a net force acting on it that has two components: one that equals its centripetal force (mac) and one that equals its tangential force (mat). The tangential force causes a change in speed (linear or angular) while the centripetal force causes a change in direction. ### Free-body diagrams and equations of motion for circular motion To solve problems on circular motion, we need to use free-body diagrams and equations of motion. A free-body diagram is a diagram that shows all the forces acting on an object. An equation of motion is an equation that relates position, velocity, acceleration, time, and/or forces for an object. The steps to solve problems on circular motion are: - Identify what type of circular motion (uniform or non-uniform) is involved. - Draw a free-body diagram for each object involved in circular motion. - Choose a suitable coordinate system (usually polar or Cartesian) and resolve all forces into components along that system. - Apply Newton's second law of motion along each component and write equations of motion. - Solve for unknown quantities using algebra or calculus as needed. ## How to solve problems on circular motion? To illustrate how to solve problems on circular motion, we will consider some common scenarios and applications of circular motion. We will also provide some examples with solutions. ### Horizontal circular motion: car on a curved road, bucket of water, etc. Horizontal circular motion occurs when an object moves in a horizontal plane along a circular path. Some examples are: - A car driving on a curved road - A bucket filled with water whirled around in a horizontal circle - A stone tied to a string swung around in a horizontal circle In these cases, we can assume that there is no friction or air resistance involved. The only force acting on the object is the tension in the string or the rod that connects it to the center of the circle. To find the tension in the string or the rod, we need to apply Newton's second law of motion along the radial direction. The net force along this direction is equal to the centripetal force. Therefore, - Fnet = Fc = T - T = mv^2/r = mω^2r where T is the tension, m is the mass of the object, v is its linear speed, ω is its angular velocity, and r is the radius of the circle. To find the speed of the object, we need to use some trigonometry and geometry. If we draw a right triangle with the string or the rod as the hypotenuse and the vertical axis as one of the legs, we can see that - sin θ = r/L - r = L sin θ where θ is the angle between the string or the rod and the vertical axis, and L is the length of the string or the rod. Using this relation, we can write - v^2/r = g tan θ - v = sqrt(rg tan θ) = sqrt(Lg sin θ tan θ) where g is the acceleration due to gravity. Alternatively, we can use conservation of energy to find the speed of the object. The total mechanical energy of the system (the object and the earth) is constant. Therefore, - Ei = Ef - mgh + 0 = 0 + 1/2 mv^2 - v = sqrt(2gh) where h is the vertical height of the object from its lowest point. ### Vertical circular motion: roller coaster, loop-the-loop, etc. Vertical circular motion occurs when an object moves in a vertical plane along a circular path. Some examples are: - A roller coaster car going through a loop-the-loop - A pilot performing a loop maneuver in an airplane - A ball attached to a string swung around in a vertical circle In these cases, we need to consider both gravity and other forces acting on the object. The net force along the radial direction is still equal to the centripetal force, but it may vary depending on the position of the object along the circle. For example, consider a roller coaster car going through a loop-the-loop as shown in Figure 6.11. The car has a mass m and moves with a speed v at any point along the loop. The loop has a radius r and is oriented vertically. Figure 6.11 A roller coaster car going through a loop-the-loop. At point A (the top of the loop), there are two forces acting on the car: gravity (mg) downward and normal force (N) from the track upward. The net force along the radial direction is - Fnet = Fc = N - mg At point B (the bottom of the loop), there are also two forces acting on the car: gravity (mg) downward and normal force (N) from the track upward. The net force along the radial direction is - Fnet = Fc = N + mg We can see that at point A, N mg. This means that at point A, gravity helps to provide centripetal force, while at point B, gravity opposes it. Therefore, N is smaller at point A than at point B. To find N at any point along the loop, we need to apply Newton's second law of motion along the radial direction and write - N - mg = mv^2/r (at point A) - N + mg = mv^2/r (at point B) To find v at any point along the loop, we need to use conservation of energy. The total mechanical energy of the system (the car and the earth) is constant. Therefore, - Ei = Ef - mgh + 1/2 mv^2 = mgh + 1/2 mv^2 where h is the vertical height of the car from its lowest point. Using this relation, we can write - v^2 = 2g(r - h) where r - h is the distance of the car from the center of the loop. ### Conical pendulum, banked road, etc. Some other examples of circular motion that involve both horizontal and vertical components are: - A conical pendulum, which is a mass attached to a string that swings around in a horizontal circle while the string makes an angle with the vertical axis - A banked road, which is a curved road that is tilted at an angle to the horizontal plane - A bicycle or a motorcycle riding on a curved road or a track In these cases, we need to consider both gravity and other forces acting on the object. The net force along the radial direction is still equal to the centripetal force, but it may have both horizontal and vertical components. Therefore, we need to resolve the forces into components along a suitable coordinate system (usually polar or Cartesian) and apply Newton's second law of motion along each component. For example, consider a conical pendulum as shown in Figure 6.12. The mass m is attached to a string of length L and swings around in a horizontal circle of radius r with a constant speed v. The string makes an angle θ with the vertical axis. Figure 6.12 A conical pendulum. There are two forces acting on the mass: gravity (mg) downward and tension (T) along the string. The net force along the radial direction is - Fnet = Fc = T sin θ The net force along the vertical direction is zero, since there is no vertical acceleration. Therefore, - 0 = T cos θ - mg Using these equations, we can find T and θ as - T = mg/cos θ - tan θ = v^2/rg We can also find v and ω using some trigonometry and geometry as - v = rω - r = L sin θ ## How to download pdf files on circular motion 12th std? If you want to learn more about circular motion 12th std, you may want to download some pdf files that contain lecture notes, slides, textbooks, reference books, sample papers, and solutions on this topic. Pdf files are convenient because they are easy to read, print, and share. They also preserve the original formatting and layout of the documents. ### Benefits of pdf files for learning circular motion 12th std Some of the benefits of pdf files for learning circular motion 12th std are: - They provide clear and concise explanations of the concepts, formulas, and methods of circular motion. - They contain diagrams, graphs, tables, and illustrations that help you visualize and understand circular motion better. - They offer examples, exercises, problems, and solutions that help you practice and test your knowledge and skills on circular motion. - They cover different levels of difficulty and complexity of circular motion problems, from basic to advanced. - They follow the latest syllabus and curriculum of circular motion 12th std. ### Sources and links to download pdf files on circular motion 12th std There are many sources and links to download pdf files on circular motion 12th std online. Some of them are: #### Lecture notes and slides on circular motion 12th std Lecture notes and slides are useful for reviewing the main points and highlights of circular motion 12th std. They are usually prepared by teachers or professors who teach this topic in schools or colleges. They may also include some tips, tricks, and shortcuts for solving circular motion problems. Some examples of lecture notes and slides on circular motion 12th std are: - [Circular Motion - The University of Sydney](http://www.physics.usyd.edu.au/helenj/Mechanics/PDF/mechanics06.pdf) This pdf file contains lecture notes on circular motion for first-year physics students at the University of Sydney. It covers the definitions, types, parameters, analysis, and problems of circular motion. It also includes some questions and answers at the end. - [Rotational Motion - Physics OpenStax](https://openstax.org/books/physics/pages/6-3-rotational-motion) This pdf file contains a section on rotational motion from an open-source physics textbook by OpenStax. It covers the rotational kinematic variables and equations, torque and lever arm, and problems and solutions on rotational motion. #### Textbooks and reference books on circular motion 12th std Textbooks and reference books are comprehensive and detailed sources of information on circular motion 12th std. They are usually written by experts or authors who have extensive knowledge and experience in this field. They may also include some additional topics or features such as appendices, glossaries, summaries, etc. Some examples of textbooks and reference books on circular motion 12th std are: - [Motion in a Plane - NCERT](https://www.ncert.nic.in/ncerts/l/keph104.pdf) This pdf file contains a chapter on motion in a plane from the NCERT textbook for class 11 physics. It covers the scalars and vectors, vector addition and subtraction, resolution of vectors, motion in a plane, relative velocity in two dimensions, projectile motion, and uniform circular motion. It also contains exercises and additional points to ponder at the end of each section. - [Motion - NCERT](https://www.ncert.nic.in/ncerts/l/iesc108.pdf) This pdf file contains a chapter on motion from the NCERT textbook for class 9 science. It covers the concepts of distance, displacement, speed, velocity, acceleration, graphical representation of motion, equations of motion by graphical method, uniform circular motion, and examples of circular motion. #### Sample papers and solutions on circular motion 12th std Sample papers and solutions are useful for practicing and testing your knowledge and skills on circular motion 12th std. They are usually prepared by teachers or experts who follow the latest exam pattern and syllabus of circular motion 12th std. They may also include some tips, tricks, and shortcuts for solving circular motion problems. Some examples of sample papers and solutions on circular motion 12th std are: - [Uniform Circular Motion Problems with Answers](https://physexams.com/lesson/uniform-circular-motion-problems-with-answers_45) This pdf file contains some problems on uniform circular motion with detailed answers. These circular motion questions are helpful for high school and college students. They cover the concepts, formulas, and methods of uniform circular motion. - [Circular Motion - Target Publications](https://www.targetpublications.org/media/catalog/product/pdf/12th-science-hsc-perfect-physics-i.pdf) This pdf file contains a chapter on circular motion from the Target Publications textbook for class 12 physics. It covers the definitions, types, parameters, analysis, and problems of circular motion. It also contains sample papers and solutions at the end of the chapter. We hope that this article has helped you to understand and learn about circular motion 12th std. If you want to download more pdf files on this topic, you can search online using keywords such as "circular motion 12th std pdf download", "circular motion 12th std notes pdf", "circular motion 12th std textbook pdf", etc. ## Conclusion Circular motion is a type of motion in which an object moves along a fixed distance from a fixed point, called the center of the circle. Circular motion can be either uniform or non-uniform. In this article, we have learned about the concepts, formulas, and methods to analyze and solve problems on circular motion. We have also provided some sources and links to download pdf files on circular motion 12th std for your convenience. Here are some frequently asked questions (FAQs) on circular motion 12th std: - Q: What is centripetal force? - A: Centripetal force is the net force acting on an object moving in a circle that makes it move towards the center of the circle. - Q: What is angular velocity? - A: Angular velocity is the rate of change of angular displacement with respect to time. It tells us how fast th