Uniform Circular Motion

Transcription

1 Uniform Circular Motion Uniform circular motion is the motion of an object in a circular path with a velocity that has a constant magnitude and a direction that is constantly changing. This is due to a force exerted on the object along the radius of the circle. This force is called the centripetal force. The acceleration of the object is the rate of change of the velocity and is given by a = v 2 /r (l) where v is the speed of the body and r is the radius of the circle. The direction of the acceleration is along the radius toward the center of the circle. This is the centripetal or radial acceleration. Newton s 2nd law for circular motion then is F = m v 2 / r (2) The force F is called the centripetal force. It is convenient in this experiment to write the velocity in quantities that can be measured. The angular velocity ω is related to velocity by the radius of the circle. That is, v = r ω (3) The angular velocity can be written in terms of frequency f, the number of revolutions per second. That is ω = 2 π f (4) Substituting Eqn. (3) and (4) into (2), the centripetal force is equal to F = (4 π 2 m r ) f 2 (5) In this experiment, a mass is rotated at the end of a string. The mass is located on a rod that can rotate and the mass is free to move along the rod. The string is fed through a pulley and connected to a force probe. The force probe measures the centripetal force. A photogate is attached to the apparatus such that it can read the period of one revolution. There is a plastic rod that extends below the fixed mass that breaks the light beam of the photogate once each revolution. The inverse of the period is the frequency of the motion. The apparatus is illustrated in Fig

2 Examining Eqn (5), if the force is plotted versus the square of the frequency, then the data should produce a straight line that should go through the origin with a slope equal to 4π 2 mr. In this experiment, part 1 is to hold the mass and radius constant and vary the frequency and measure the force. In part 2, the radius is constant and different masses are added. However, the frequency will change due to the different load on the electric motor. Therefore, the force, mass, and frequency are measured at a fixed radius and the force is calculated using Eqn (5). APPARATUS: 1.Centripetal force apparatus 2. Photogate 3. Supporting stand 4. Force Probe 5. Logger Pro V Power Supply Procedures: 1. The Centripetal Force Apparatus should already be assembled. There is a switch on the force probe. Make sure the switch is set to the 10 N setting. Check to make sure the force probe and the photogate are plugged into Logger Pro and Logger Pro is plugged into an electrical outlet. Connect your laptop to Logger Pro and open the program. Go to File/Open/Probe & Sensors/Photogates/strobe. The program should sense that a force probe is also connected and you should click the box accordingly. The table should display column heading of Time, Gate Status, and Frequency. Next, add the Force column by navigating to Experiement/Set up Sensors/Show all Interfaces. Click on Channel 1 and add the Dual Range Force Sensor. We will be using the Frequency and Force columns. 35

3 Part 1- Force versus frequency 2. The rotating mass is the mass that is attached to the string. The mass is the sum of the mass of the weights plus the mass of the thumbscrew (4 g). The fixed mass is placed at the same position on the other side. It is important that the rod be balanced so that it will not wobble when rotated. The radius can be read on the scale below the mass. The string may be slack so pull the mass out to read the radius. Add a 20-gram mass to both sides of the apparatus and record the mass and the radius in the data table. 3. Set the voltage to 2 V. When the apparatus starts rotating, push collect. Collect data for 3 seconds and push stop. The tables should be filled with data. The force column is not so constant and does oscillate some. To find the average value, go to the Data menu and open new calculated column. Name the column Newtons and go to function/statistics/mean and then go to variable and choose force. The average value should appear in next column. Do the same for average frequency and name it Hertz. Record the frequency and force in data table. Repeat for voltages 3 thru 10 V at 1 V intervals. 4. Plot the Force versus the frequency squared and fit the curve to a straight line. Record the slope in data table. Compare the experimental slope with the theoretical value (4π 2 mr). Part 2 - Force versus mass 5. Set the power supply to 5 V and keep it constant throughout this part. Place 10 g on both sides of the apparatus. The rotating mass now totals 14 g. Start the apparatus rotating and measure the frequency and force. Record the mass and the radius in the data table. 6. Increase mass on each side of rod by 5 g and repeat the measurements. Continue in 5 g increments to a maximum of 34 g. Be sure the radii of both masses remain equal throughout experiment. 7. Using an Excel spreadsheet, calculate the force using Eqn (5) and find a percent error. 36

4 Data Table: Part 1 - Force versus Frequency Mass = Experimental Slope = Percent error = Radius= Theoretical Slope Frequency Frequency squared Force 2V 3V 4V 5V 6V 7V 8V 9V 10V Part 2 - Force Versus Mass Radius= Mass (kg) Measured Force Frequency Calculated Force Error Show a sample calculation below. 37

5 Questions: 1) How would the motion of the mass be affected if the apparatus was rotated such that the plane of the circle was vertical? 2) Because the Earth rotates about its axis, you will have an apparent weight that is slightly less at the equator than at the poles. Compare the normal force of an 80 kg person at the equator and at the north pole. n pole = n equator = 38