Uniform circular motion – Another specific example of 2D motion

1. Uniform circular motion – Another specific example of 2D motion An object that is traveling in a circular path is moving in two Cartesian directions (x and y for example) simultaneously. What do you know about an object that is moving along a circular path? y v v The radius is constant and the speed is constant. a a Is the object accelerating? v Yes! – the direction of the velocity changes. a a x What is the direction of the acceleration? r v The acceleration is directed towards the center of the circular path. This is called centripetal acceleration. a v The centripetal acceleration is what keeps the object moving in a circular path. What would happen to the object moving in a circular path if the centripetal acceleration was instantaneously reduced to zero? The object would move in the direction of the velocity at the instant the acceleration was reduced to zero. This would result in a straight line path that is tangent to the circular path at that instant.

2. The position of an object will change as it moves along a circular path. The position vectors are defined relative to the center of the circular path for this case. This will result in a change in position defined by Dr. The radius of the path does not change, therefore Dr corresponds to a directional change (Dq) only. Dq The velocity will also change. Since there is no acceleration along the direction of the velocity you again have only a directional change (Dq). Dq Notice that Dr is in the direction of the velocity (tangent to the path), Dv is in the direction of the acceleration (perpendicular to the path) and since r and v are perpendicular both change through the same angle Dq. Ds - Arc length Using the definition for arc length we can relate Dr to r through Dq. (The arc length will be linear over a small angular change. Ideally we would be using the differential form of these quantities.) We can apply this definition to velocity as well.

3. Using the relationship we obtained from an analysis of the position and velocity vectors we can obtain an expression for the centripetal acceleration. (This analysis has been simplified and is therefore not the most accurate method that can be used. We should be using the differential form of each of these quantities.) v In the limit that Dt -> 0 Centripetal acceleration is always directed towards the center of curvature. (Towards the center of the circular arc.) Centripetal acceleration

4. How long does it take a particle to make one complete revolution? For circular motion the speed of the particle is constant. s is used here to represent the distance along the path of the particle, also called the arc length between two points on a curved path. The arc length is related to the radius of the circular path and the angular change in position. (r is constant for a circle.) 0 q = 2p for a complete circle T is used to represent the period (time to complete one revolution). 0 q/T is the angular velocity (w) This give the time to complete one full revolution for a specified radius and speed. Notice that this is the circumference of the circular path.

5. Example: What is the centripetal acceleration of a speck of dust on the outside edge of a DVD? The DVD has a radius of 6 cm and it takes the speck of dust 0.29 s to go around once. Physics Explore the Universe R Towards the center

6. Circular Motion Using a Radial and Tangential Coordinate System It is sometimes more convenient to define a new coordinate system when discussing curvilinear motion. The radial direction is defined to be positive outward from the center of curvature. The tangential direction is defined to be positive in the direction you measure a positive angle. - Tangential unit vector - Radial unit vector Notice that the unit vector direction changes as you move along the circular path. Centripetal acceleration is often described using this coordinate system (or Polar Coordinates). The total acceleration of an object using this coordinate system would contain both radial and tangential components.

7. In general the acceleration and velocity will have a radial and tangential component. The velocity for uniform circular motion is constant along the tangential direction. If the velocity is constant there is no tangential component of the acceleration, but there is still a radial component of the acceleration directed in the negative r direction. The centripetal acceleration is the radial component of the acceleration and will only change the direction of the velocity. If there is a tangential component of the acceleration it will cause a change in the magnitude of the tangential component of the velocity.

8. The apparatus pictured is positioned on a horizontal surface. A small ball bearing (in container at lower left of picture) is blown through the spiral hose, emerging at the right side and moving downward (in the picture), toward one of the five aluminum tube targets. Which of the targets will it hit? 1 2 3 4 5

9. A pool ball is rolled clockwise around the inside of the circular band. Which path will the ball follow? 1 2 3