Circular Motion. We will deal with this in more detail in the Chapter on rotation!


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1 Circular Motion I. Circular Motion and Polar Coordinates 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 2dimensional. (X B,Y B ) r (X A,Y A ) B If we use polar coordinates, the radius is constant and only the angle theta changes. This simplifies the system to a 1dimensional problem and makes the math simpler. We will deal with this in more detail in the Chapter on rotation! B. Tangential Velocity By its definition in terms of the derivative of the position vector, the velocity vector is Tangent to the Curve every point on the ball's path. Thus, we call it the Tangential velocity. This is the same velocity that we dealt with in 1dimensional and projectile motion problems.
2 For a rigid body composed of many particles traveling in circles of different radii, it is convenient to also define another type of velocity (angular velocity) based upon polar coordinates. Thus, we need to be careful in our terminology when discussing velocity to ensure the reader knows which velocity we are talking about. C. Acceleration 1. Acceleration is defined as the Time Rate of Change of the Velocity Vector. A Velocity Vector has two parts: and. If either part changes then the object is undergoing Acceleration. 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 Tangential acceleration is the acceleration an object feels due to a change in the object s. The magnitude of the tangential acceleration is usually either specified in the problem statement or found using trigonometry.
3 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 of Tangential Acceleration as Velocity vector if object is Speeding Up of Velocity vector if object is Slowing Down Speeding Up Slowing Down 3. Centripetal (Radial) Acceleration Centripetal acceleration is due to the change in the of the Velocity Vector Centripetal means Center Seeking This tells you that the centripetal acceleration always points to the Center of the Circle. It is always Perpendicular to the Tangential Acceleration.
4 Any object traveling in a Path MUST HAVE Centripetal Acceleration. Furthermore, notice that the Centripetal Acceleration is always Perpendicular to the Tangential Velocity Vector. 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: 2 v r r a 2 This is a very useful formula for solving problems. Students in trig. based physics courses can either derive the equation using trigonometry or just memorize the equation. However, the equation can be derived easily using the basic definition of acceleration using polar coordinates once a student takes Calculus.
5 4. Total Acceleration Tangential and Centripetal Acceleration are our acceleration components in polar coordinates. The total acceleration of an object traveling in a circle is thus the vector sum of the tangential acceleration and the centripetal acceleration. 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 m. What is the total acceleration on the car when it has slowed to 20.0 m/s as shown below:
6 II. Uniform Circular Motion An object that is traveling in a circle at constant speed is said to be traveling in uniform circular motion. This is just a special case of circular motion where the object has no tangential acceleration. It does have centripetal acceleration. 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
7 III. General Curvelinear Motion In A Plane As our concept question shows, any curvelinear 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 magnitude. 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 There Is New Physics!!
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