Dynamics Circular Motion, Part 1

Centripetal Acceleration and Force In the previous unit we learned that when an object moves through a curved path there is a component of acceleration perpendicular to the tangential velocity vector. This acceleration is known as centripetal acceleration, ac . Acceleration is caused by an unbalanced force. Therefore, centripetal acceleration must be caused by an unbalanced centripetal force, Fc . When an object turns it does so due to a centripetal force, which is directed toward the center of the turn and is tangential to velocity. In uniform circular motion objects move at a constant speed in perfect circles. This means that there is NO tangential acceleration or force. Since the object is turning there is centripetal acceleration and centripetal force. These vectors are directed toward the exact center of the circular path in uniform circular motion. ac Fc v

Centripetal Acceleration and Force Fc v Just as with linear motion, acceleration and force in circular motion are related with Newton’s 2nd law. Centripetal force is NOT necessarily due to a single force. Centripetal force is the net force force acting toward the center of the circle. In circular motion problems centripetal force is the sum of the forces and components of forces that are directed perpendicular to the tangential velocity of the object. This means that Fc replaces F in the sum of forces equation when solving circular motion problems. IMPORTANT: The vector Fc is never shown in free body diagrams. It is the sum of the applied forces that act on the object to make it turn. Only the actual applied forces should be in the diagram.

Practice with FBD’s and Summing Centripetal Force The next series of slides will show several situations involving circular motion in order to practice summing forces. a. Draw a free body diagram for each example. Free body diagrams are the same in all force problems. There is no difference when circular motion is involved. Just keep in mind that Fc is NOT added to the diagram. b. Write the relevant sum of forces equation(s) for each example. If the problem involves a circle (or even part of a circle) use Fc instead of F to start the equation. Forces pointing toward the center of a circle are positive Forces pointing away from center are negative

Vertical Circles and Loops These problems usually involve a circular track, such as a rollercoaster, or a ball tied to a string that is swung in a vertical circle. NOTE: These motions are not uniform circular motion, which requires constant speed. In vertical circles objects slow as they move upward reaching a minimum speed at the top of the loop, and then they speed up moving downward reaching a maximum speed at the bottom of the loop. During the motion there is a tangential acceleration and force, which are responsible for changing the objects speed during the motion. However, at two key instants the tangential acceleration and force are both zero. When this occurs there is only a centripetal force and acceleration present, and at these two points vertical circles solve in the same manner as a uniform circular motion problem. The two key instantaneous points that we are able to solve easily are the very top of the circular path and at the very bottom. Solving other locations will be shown in the rotation unit.

Example 1 +N +T −Fg Bottom of vertical loops Ball, car, or rollercoaster Pendulum Ball spun at end of string −Fg +N +T

Example 2 −T −N +N +Fg Top of vertical loops Ball, car, or rollercoaster Rollercoaster loop Ball spun at end of string +Fg −N −T +N

Example 3 −T −N +N +Fg +Fg Special case at the top of vertical loops There is a special case that occurs only at the top of vertical loops. At a specific speed the object experiencing circular motion will pass through the highest point of the loop and exert no force on the surface that it is moving along or the string it is attached to. At this special speed the normal force in the two left hand scenarios will become zero, and the tension in the right scenario will become zero. When normal force or tension are zero the object is apparently weightless. This special case will be indicated in problems as shown in the next slide.

Example 3 −T −N +N +Fg +Fg Special case at the top of vertical loops The language indicating that you should set N or T to zero varies, but it will be words similar to the examples shown below each scenario. +Fg −N −T +Fg +N Maximum speed the object can have without leaving the track. Or Feel weightless The minimum speed the object can have and still complete a loop.

Example 3 +Fg +Fg +Fg Special case at the top of vertical loops Maximum speed the object can have without leaving the track. Or Feel weightless The minimum speed the object can have and still complete a loop. Now each scenario is essentially the same and solves the same.

Example 4 N f T Fg Horizontal circular motion Attached to string Object on a rotating disk Car in a flat turn frictionless surface and not slipping and not skidding f N Fg T