12d-Centripetal Force Lab 1-17-09 - 1 - CENTRIPETAL FORCE Introduction The purpose of this lab is to use Newton’s 2nd Law to predict the dynamic centripetal force on a rotating mass based on the measurement of the mass (m), radius of rotation (r), and the period of rotation (T). This force will then be measured statically and compared to the prediction of Newton’s 2nd Law. Equipment Computer with Logger Pro SW Slotted weights Digital Scale Vernier Lab Pro Interface Ring stand Meter stick Photogate & connecting wires Sting with hook (paper clip) Clamps (2) Weight hanger Clamp, 90o swivel ½” x 1-1/2 card Theory When an object of mass m is made to travel in a circular motion at constant linear speed v it is undergoing centripetal acceleration. A centripetal or center-seeking force is said to be acting upon the object. The force and acceleration are related by Newton’s 2nd law, F = ma (1) with the centripetal acceleration in the form:12d-Centripetal Force Lab 1-17-09 - 2 - a = v2/r (2) where r is the radius of the circular path of the object. Therefore the expression for the centripetal force can be written as: F = mv2/r. (3) The magnitude of the linear velocity of the rotating object equals the circumference of the circular path of the object’s motion divided by the period of the motion T: v = 2π r/T F = m 4π2r /T2 (4) In the first portion of this experiment, you will dynamically determine the values of the period T and the radius r by rotating a support bar with a mass hanging from it and timing multiple revolutions. A spring attached to the mass supplies the centripetal force. You will need the values of the period T and the radius r in order to compute the value of the centripetal force. Putting these values into equation (4) above gives an experimental value for centripetal force. (The radius is marked on the apparatus.) In the second portion of the experiment, the apparatus will not be rotating and the value of the centripetal force will be determined statically. A string will be attached to the mass, and the string will pass over a pulley. The spring will be stretched by weights on a weight hanger attached to the string running over the pulley. (The pulley is shown in the photo, but the string and weights are not.) Equipment Procedures Different versions of the Centripetal Force apparatus: The older version of the apparatus is shown in the picture on the first page of this lab and has a solid metal base. The second and third versions have three leveling adjustment screws in the wooden base. The second the screws themselves contact the table while the third version has black plastic feet end the bottom of the leveling screws that make contact with the surface of the table. This difference in the various apparatus is at the heart of the trade off between level operation and stability. Centripetal Force 1st Ed. 2nd Ed. 3rd Ed. Platform Metal Wood Wood Leveling Screws None Metal screws Metal Screws w/ plastic feet Clamping to Table Not needed Square on corner of lab table. Back off leveling screws. Diagonal on corner of lab table so leveling screws don’t touch the table. Radius Marker Fixed and labeled Adjustable and unlabeled Adjustable and unlabeled Rotation Turn from above Knurled area on support rod. Knurled area on support rod.12d-Centripetal Force Lab 1-17-09 - 3 - Experiment File: Under the File menu select the Open menu item. The Experiments folder will appear, double click on the Probes and Sensors folder, then double click on the Photogate folder and then, finally, double click on the Pendulum Timer file. Flag Set Up: Install a “flag” on the top of the rotating mass and place a photogate in the circular path that the flag will follow so that the flag will break the beam once per revolution. This will not be the usual flag that was used in previous labs. A rectangular piece of thin cardboard (1/2” by 1-1/2”) will do. The photogate in this program will function as an event timer. The event will be “the blocking of the beam” by the flag. The width of the flag doesn’t matter because the program is counting the time between the beginning of every second blocking event. Radius Adjustment: Detach the spring from the rotating mass. Loosen the screw in the top of the support column through which the rotating arm passes. Adjust the position of the rotating arm so that the mass hangs directly over the desired radius marker. Make sure there is enough clearance (about 2 millimeters) between the pointed tip on the bottom of the mass and the radius marker. Rebalancing Procedure - Rotation Radius: Each time the rotation radius is changed the rotating arm needs to be rebalanced. With the support column screw still loosened, move the counter weight at the other end of the rotating arm either in or out until the rotating arm Rotating mass: m Counter Weight Rotating Arm Pulley Support Column Rotation Radius Markers Support Column Screw (not shown) Flag12d-Centripetal Force Lab 1-17-09 - 4 - will balance with the screw loosened. Once the arm is balanced tighten the support column screw securely. Flag Alignment Test: At this time (with the spring still detached), turn the rotating arm SLOWLY to ensure that the flag attached to the top of the rotating mass interrupts the photogate beam correctly. This flag will always rotate in the same horizontal plane so it is the vertical height of the photogate that needs to be adjusted so that the flag passes through the midpoint of the beam path. Re-attach the spring to the mass. Experimental Procedure Repeat both the Dynamic and Static portions of the experiment for a total of five different radii. Data Collection (Dynamic): Rotate the apparatus at a constant speed by turning the top of the shaft with your fingers so that the pointed tip of the hanging mass continually passes squarely over the chosen radius mark. Assume this method gives a constant speed. Use the mouse to click the Collect button on the computer screen. Click the Stop button after about 20 revolutions. Use Statistics from the Analyze Menu to average the times. NOTE: The gate time measured is two times the period T. This is due to the nature of the pendulum that passes through the photogate twice per period while our rotating mass passes through the photogate once per period. All this means is that you need to divide your average gate time by 2 to get the