Rotational Motion AP Physics1

Introduction to Rotational Motion

Speaking of rotational motion, you may think of merry-go-round, a fan or even our rotating earth. Rotational motion is all about taking an object and spinning it. The object follows many rules while spinning just like the way they do when moving on a straight line. You may think it’s easy to get confused by those rules, but being analogous to physics of linear motion actually makes the rules of rotational motion very easy to memorize.

In this course , we are going to discuss some basic terms and a set of equations in rotational motion. This  course will go through all you need to know about rotation and talk about rotation-related questions in AP® Physics and how those equations apply to solve real test questions.

Basic Elements in Rotation

Center of Mass

To tackle rotation-related questions, we should first expand the moving object from a point to a real object with shape and scale. In linear motion, we usually treat objects as if they are single particles, and we assume that all the forces are exerted only on a single point that represents the object. But how can we be sure? Why does that point summarize and represent the whole object?

If you have ever balanced a pen on your finger, you should be very familiar that you could always find a specific point to keep the pen from falling. The position of the point varies from pen to pen and are not always in the middle. That certain point is called center of mass. In other words, the center of mass is the point where all the mass of the object could be considered to be concentrated.

Pivot Point

A pivot point is a center of rotation. It could be the center of mass or any point that you select. Force exerted on this point will not contribute to the rotation.

Torque

Torque is the measure of a force’s effectiveness as making an object accelerate rotationally. Still don’t know what a torque is? To understand torque, first, let’s recall a time that you use a wrench to loosen a bolt.

Angular Displacement, Velocity and Acceleration

Objectives:

• Relate radians to degrees.

• Define average angular velocity.

• Define instantaneous angular velocity.

• Define average angular acceleration.

• Define instantaneous angular acceleration.

• Distinguish linear velocity from angular velocity.

• Distinguish linear acceleration from angular acceleration.

• Determine the linear and angular velocities and accelerations in a physical situation given sufficient information

• Angular Velocity and Angular Acceleration

  If the object is already spinning, what are variables to characterize its movement? Just like velocity and acceleration in linear motion, angular velocity and angular acceleration depict the movement of an object or of a particular point on the object. Angular velocity depicts the rate an object is spinning. It equals to the angle turned divided by time, ​. The direction of angular velocity is perpendicular to the rotational plane, determined by the right-hand rule. Angular acceleration is the change rate of ω, denoted as α.

  Moment of Inertia

  Also known as rotational inertia, the moment of inertia is used to characterize the tendency of an object to continue rotating before a torque is exerted on it. To better understand inertia, let’s compare inertia to mass. Mass is the tendency of an object to resist changes in its motion. To make an object move, we add a force on it. To make an object rotate, we exert a torque. While F = ma, torque equals to moment of inertia times angular acceleration.

  
  

  So what determine the moment of inertia? Let’s first look at an example in figure skating. If the athlete wants to speed up while spinning, they draw in their arms and legs. Why will this do? Because inertia is related to the distribution of mass. The further an object is from the pivot point or the axis, the harder it is to make it spin. So when the athletes fold their arms, they are reducing their rotational inertia, therefore reduce the torque required to make them spin and elevates the angular

• Angular Velocity and Angular Acceleration

  If the object is already spinning, what are variables to characterize its movement? Just like velocity and acceleration in linear motion, angular velocity and angular acceleration depict the movement of an object or of a particular point on the object. Angular velocity depicts the rate an object is spinning. It equals to the angle turned divided by time, notated as \omega =\dfrac { \Delta \Phi }{ \Delta t }ω=ΔtΔΦ​. The direction of angular velocity is perpendicular to the rotational plane, determined by the right-hand rule. Angular acceleration is the change rate of ω, denoted as α.

  Moment of Inertia

  Also known as rotational inertia, the moment of inertia is used to characterize the tendency of an object to continue rotating before a torque is exerted on it. To better understand inertia, let’s compare inertia to mass. Mass is the tendency of an object to resist changes in its motion. To make an object move, we add a force on it. To make an object rotate, we exert a torque. While F = ma, torque equals to moment of inertia times angular acceleration. Rotational inertia is denoted as I.

  \tau =I\alphaτ=Iα

  So what determine the moment of inertia? Let’s first look at an example in figure skating. If the athlete wants to speed up while spinning, they draw in their arms and legs. Why will this do? Because inertia is related to the distribution of mass. The further an object is from the pivot point or the axis, the harder it is to make it spin. So when the athletes fold their arms, they are reducing their rotational inertia, therefore reduce the torque required to make them spin and elevates the angular velocity.

  
  

  About Rotational Equilibrium

  After getting familiar with those terms in rotation, we can see how those terms apply in solving problems. First, we would like to determine when an object can reach the equilibrium. The translational equilibrium is reached when the sum of force acting on the object is zero. Similarly, the rotational equilibrium is reached when the sum of torque acting on the object is zero. Be careful, being in a rotational equilibrium state does not necessarily mean the object is not rotating, it could also be rotating around its center of mass at a constant speed. If the object is totally at rest, it is said to be in a static equilibrium.

  \sum { \tau } =0∑τ=0

  
  

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Angular Kinematics and Rolling Motion

Objectives:

• Work with the four kinematics equations for constant angular acceleration to determine missing information.

• Solve problems involving rolling motion where no slippage occurs.

  Torque

  
  

• Objectives:

  • Qualitatively describe the concept of torque.

  • State the mathematical relationship between torque, force, direction and distance of application.

  • Given sufficient information, determine the torque being applied in a variety of physical situations.

  
  

First part of course ie Rotaional inertia sets up a stage for the study of rotational kjnetic energy and angular momentum by developing understanding about rotational inertia of an object or system.

Its better if students have a basic understanding that mass describes inertia in translation motion.

The student will learn to plan data collection strategies designesd to establish that torque , angular velocity , angular acceleration and angular momentum can be predicted accurately when the variables are being treated clockwise or counterclockwise with respect to well defined axis of rotation and refine the research question based on data. This will be done in two ways

1. Describing a model of rotational system

2. Using that model to analyse a situation in which angular momentum changes due to interaction with other object or system.

Objectives:

• Define moment of inertia.

• Calculate the moment of inertia in simple systems.

• State the relationship between torque, moment of inertia, and angular acceleration.

• Systematically solve numerical problems involving rotation, including problems with pulleys.

In the second part of rotational kinetic energy student will learn to calculate total energy of a rolling object using both translational and rotational kinetic energies.

Objectives:

• Use the equations for translational and rotational kinetic energy to find the total kinetic energy of a rolling object.

• Determine the rotational kinetic energy and translational kinetic energy for an object rolling down an incline.

Student is able to calculate changes in kinetic energy and potential energy of a system using information from representations of that system.

The student is able to make claims about the interaction between system and its envronment in which environment exert force on the system , thus doing work on the system and changing the energy of the system (Kinetic energy plus potential energy)

Student will also learn to apply work energy theorem to examine how torque can do work in changing rotational kinetic energy of an object or system.

In the third part ie momentum and conservation of momentum student will learn how linear momentum and conservation of linear momentum analogous to analogous to angular momentum and conservation of angular momentum. This part gives strong emphasis on predictions on the behaviour of rotational collisions by the same process which are used to analyze linear colision situations using analogy between impuse and angular impulse.

Objectives:

• State Newton’s Second Law for Rotation.

• State the Law of Conservation of Angular Momentum.

• Apply the Law of Conservation of Angular Momentum in a variety of physical situations.

• Describe the vector nature of angular quantities – which way are they oriented?

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