I. Circular Motion and Polar Coordinates Lesson 8 Kinematics V - Circular Motion A. Consider the motion of ball on a circle from point A to point B as shown below. We could describe the path of the ball in Cartesian coordinates or by polar coordinates. In Cartesian coordinate system, we see that both coordinates change!! This makes the problem 2- dimensional. (X B,Y B ) r (X A,Y A ) θ B If we use polar coordinates, the radius is constant and only the angle theta changes. This simplifies the system to a 1-dimensional problem and makes the math simpler. We will deal with this in more detail in Chapter 9 when we study rotation! B. Tangential Velocity By its definition in terms of the derivative of the position vector, the velocity vector is to the at every point on the ball's path. Thus, we call it the velocity. This is the same velocity that we dealt with in 1-dimensional and projectile motion problems. We will see in Chapter 9 that for rigid bodies composed of many particles traveling in circles of different radii that it is convenient to define another type of velocity (angular velocity) based upon polar coordinates.
C. Acceleration 1. Acceleration is defined as the of of the. A has two parts: ( ) and. If either part changes then the object is undergoing. Thus, it is often convenient to break the acceleration into components based upon the change in speed (magnitude) or direction of the velocity vector instead of x and y directions. This is again an example of using polar coordinates to represent the motion. 2. Tangential Acceleration This is the acceleration an object feels due to a change in the of the. Magnitude Formula The magnitude of the tangential acceleration is rarely found using this formula. It is usually specified in the problem statement or found using trigonometry.
The tangential acceleration is the only acceleration possible for straight line motion. We can use this to help us find the direction of the acceleration vector. Direction as if of if Speeding Up Slowing Down Using polar coordinates, a physicist can combine the information in the previous statements and write the following single formula for the tangential acceleration of an object in circular motion: 3. Centripetal ( ) Acceleration Centripetal means. This tells you that the centripetal acceleration always points to the of the. It is therefore to the.
Centripetal acceleration is due to the change in the of the. Any object traveling in a MUST HAVE. Furthermore, notice that the is always to the. This is why the moon can be accelerating toward the Earth while not moving toward the Earth!! The magnitude of the centripetal acceleration vector can be found by the formula: This is a very useful formula for solving problems and can be derived directly from the definition of acceleration using Calculus. This derivation is usually reserved for students in either Engineering Principles I (Dynamics) or the Junior Level Mechanics class for Physics and Engineering Physics Majors. You will probably prefer just to memorize the equation.
4. Total Acceleration The total acceleration of an object traveling in a circle is the of the and the. Example: A car is slowing down at a rate of 6.00 m/s 2 while traveling counter clockwise on a circular track of radius 100.0 m. What is the total acceleration on the car when it has slowed to 20.0 m/s as shown below:
II. Uniform Circular Motion An object that is traveling in a circle at is said to be traveling in. This is just a special case of circular motion where the object has. It does have.
Concept Question Consider the case of projectile motion from the last lesson: A cannon ball is fired out of a cannon and follows a parabolic path before hitting the ground. What type(s) of acceleration does the cannon ball have during its flight at point X? X A. Tangential Acceleration B. Centripetal Acceleration C. Both Tangential and Centripetal Acceleration D. Neither Centripetal or Tangential Acceleration
III. General Curve-linear Motion In A Plane As our concept question shows, any curve-linear motion can be seen at every instant as circular motion a circle whose radius is the radius of curvature of the trajectory at that particular point. In the case of straight line motion, the radius of curvature is infinity so their is no centripetal acceleration. In the case of circular motion, the radius of curvature is constant! So why didn't we treat projectile motion using the concepts of centripetal and tangential acceleration? Because it makes the math harder to perform!! In Cartesian coordinates, the acceleration has only one component (vertical) and it is constant in magnitue. Thus, we can use the kinematic equations. In polar form, both the tangential and centripetal acceleration components vary in direction and magnitude. Thus, we couldn't use the kinematic equations with either component. We Use Different Coordinate Systems and Define New Quantities To Make The Math Simpler For Solving Problems. This Doesn't Mean That Their Is New Physics!!