* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download LAB X: Title
Survey
Document related concepts
Classical mechanics wikipedia , lookup
Jerk (physics) wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Fundamental interaction wikipedia , lookup
Nuclear force wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Coriolis force wikipedia , lookup
Fictitious force wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Centrifugal force wikipedia , lookup
Transcript
LPC Physics Centripetal Force Centripetal Force Purpose: The purpose of this lab is to measure centripetal force and compare it with predicted values, and to demonstrate that a centripetal force acts to hold an object in a circular orbit. Equipment: Centripetal Force Apparatus Hooked Mass Set Stopwatch Balance Carpenter’s Level Meter Stick Photogate, LabPro Theory: The change of velocity of an object is, by definition, an acceleration. If an object is to accelerate there must be a force acting on the object which points in the direction of the acceleration (in the direction of the velocity change). Velocity is a vector quantity, meaning it has both a magnitude and a direction. A change in velocity (that is, an acceleration) may be due to a change in the magnitude (faster or slower), a change in direction, or both. If an object is moving in a circle, it necessarily undergoes an acceleration as can be seen by its constant change of direction. 1 of 9 LPC Physics Centripetal Force vi ∆v = vf - vi vf Figure 1 Subtraction of vi from vf gives ∆v, or acceleration In an exaggerated example, an object traveling counterclockwise in a circle will change the direction of its velocity as it moves from its starting point at 12 o’clock to its final point at 9 o’clock. The change in velocity, ∆v = vf - vi is shown pictorially above as pointing directly toward the center of the circle. This means that the object is accelerating toward the center of the circle, which means there must be a force pointing toward the center of the circle. This force is known as a centripetal force. Centripetal means “center seeking”, and should not be confused with the more commonly known socalled centrifugal (“center fleeing”) force which is actually not a force at all. A note on vector subtraction: Vectors are added “tip to tail”, meaning that the end (with no arrow) of the second vector is placed at the tip (with the arrow) of the first vector and the resultant vector is drawn from the end of the first vector to the tip of the second. Just like with “regular” numbers, it doesn’t matter which is the first and which is the second vector. v1 + v2 : v1 v2 + v1: v2 v2 vf v1 v1 v2 vf Vectors are subtracted by placing the ends (no arrow) of the two vectors together, and drawing the resultant vector from the tip of the second to the tip of the first. Just like with “regular” numbers, it does matter which is the first vector and which is the second vector. Transposing the two results in a negative answer. v1 - v2: v2 - v1: v1 vf v1 vf v2 v2 If an object is to travel in a circle, the centripetal force must be equal to Fcentripetal mv 2 = r where m = the mass of the object v = the velocity of the object r = the radius of the circle 2 of 9 Eq. 1 LPC Physics Centripetal Force From this equation you can see that to travel in a circle of smaller radius requires a larger centripetal force, as does a larger mass, and a higher velocity. So what happens if an object is attempting to trace a circle of radius r1 a constant velocity, and the centripetal force that exists is not large enough to meet the condition in Eq. 1? velocity At any point around a circle, the instantaneous velocity is pointed tangent to the circle. Thus, an object’s instantaneous momentum points tangent to the circle, and upon the removal of a force, the object will continue to move in the direction of its momentum. r1 r2 If the centripetal force is not great enough to keep the object in a circle of radius r1, the object’s tangential momentum will Path of object carry it outward until it reaches the radius that will drift make Eq. 1 true. In essence, the object drifts outward to “find” the r1 appropriate radius. If the force were removed entirely, the object would travel in a straight line. Because in this case there still exists a force, the path of the object is curved, and eventually (at the right radius) becomes a circle again. Figure 2 An object will drift to a circle of larger radius if the centripetal force acting on it is too small Conversely, if the force applied is greater than that required to push an object in a circle of radius r1, the object will be pushed inward to a smaller radius that meets the condition of Eq. 1. But wait, you might say, Eq. 1 includes velocity as well as radius. And we all know that a force may change an object’s velocity. So why mightn’t, in the first case above, the object just decrease its velocity rather than move to a larger radius? The velocity won’t change for the same reason that gravity will never change the horizontal velocity of a projectile…a force must be at least partially in the object’s direction of motion to change its velocity. If a force acts perpendicularly to an object’s direction of motion, the velocity will neither increase nor decrease, but the force will change the direction of motion. It is very important to remember (even more to realize for the first time) that Centripetal Force is not a force in the same way that gravity, rope tension, the normal force, etc. are forces. The centripetal force is simply a label for any of these other forces that happen to be causing an object to move in a circle. In the case of the Earth orbiting the Sun (in a mostly circular fashion), we don’t label the force of gravity and the centripetal force acting on the earth to push it in a circle. We say that the force of gravity acts on the Earth, and acts as a centripetal force to push the Earth in a circle. 3 of 9 LPC Physics Centripetal Force The Earth-Sun System Wrong! Fcentripetal Fgravity Sun Earth Ftotal = Fcentripetal + Fgravity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Right! Fcentripetal = Fgravity Sun Earth Ftotal = Fgravity which is acting as Fcentripetal Figure 3 The exerted on the Earth by the Sun: Right and Wrong interpretations The same could be said about a ball on a string being swung in a circle. Because the ball is traveling in a circle, there is a centripetal force acting on it. The force that is causing the centripetal acceleration is the force of rope tension. A Ball on a String Right Wrong ΣF = Frope tension which is acting as Fcentripetal ΣF = Fcentripetal + Frope tension Ftotal Ftotal Frope tension Fcentripetal Frope tension is Fcentripetal Figure 4 Centripetal forces on a tennis ball: Right and Wrong interpretations And this is the way it will go every time you have an object that travels in a circle. The first step is recognizing the circular motion, and realizing that a force (centripetal force) is necessary to push the object in a circle. The next step is determining what type of force is acting as the centripetal force. Any force you can think of can act as a centripetal force. 4 of 9 LPC Physics Centripetal Force So what is this lab all about? You’re going to determine the centripetal force necessary for an object of known mass to travel in a circle of a certain radius at a calculated velocity. If you’ve been to a carnival (or to Camp Snoopy™ at the Mall of America), you may have seen the giant swings. About 50 people are lifted 20 feet off the ground and swung in a large loop around a central axis. As the angular velocity increases, the rider feels “pushed”outward until the chains are no longer vertical but at some small angle. At this point, it is the horizontal component of the rope tension that acts as the centripetal force. ω Frope tension Frope tension, vertical Frope tension, horizontal = Fcentripetal Figure 5 A close-up view of forces on a rotating swing Now imagine that a spring were attached to the swing chair such that when stationary the chair is pulled in toward the central axis. Now when the chairs are spun, the spring will stretch until the chair hangs vertically. In this instance the spring force acts as the centripetal force. ω ω=0 Fcentripetal = Fspring Figure 6 Because the spring force is proportional only to the distance the spring is stretched, once we’ve stopped spinning we can determine what was the centripetal force on the swing by applying enough force to stretch the spring the same distance. 5 of 9 LPC Physics Centripetal Force ω=0 Fappliedl = Fspring Figure 7 You will be performing this same experiment on a smaller scale. You are provided with a rotating arm that has a hanging mass on one end and a counter-weight on the other. The index on the platform (thin rod) provides a means of measuring the radius of rotation. A photogate or stopwatch may be used to determine the velocity of the rotating mass. From this you will calculate a “theoretical” centripetal force. Once the spinning has been completed, you will stretch the spring by means of hooked masses strung over the provided pulley. This will determine the “experimental” centripetal force. You shall then compare the two.1 Experiment: 1. Level the base of the centripetal force apparatus. 2. Determine the mass, m1, of the hanging mass. 3. Connect the AC adapter to the LabPro by inserting the round plug on the 6-volt power supply into the side of the interface. Shortly after plugging the power supply into the outlet, the interface will run through a self-test. You will hear a series of beeps and blinking lights (red, yellow, then green) indicating a successful startup. 4. Attach the LabPro to the computer using the USB cable that is Velcro-ed to the side of the computer box (do not unplug the USB cable from the computer!). The LabPro computer connection is located on the right side of the interface. Slide the door on the computer connection to the right and plug the square end of the USB cable into the LabPro USB connection. 5. Connect the Photogate to the DIG/SONIC1 port of the LabPro. If you are using a one-piece Photogate, a PASCO or very old Vernier Photogate, you will need to use the digital adapter. If you are using a newer Vernier (with removable cable), simply remove the cable with the Phono plug, and connect the Photogate Cable with a British-Telecom plug on one end. 6. Open the file centripetal_force.cmbl (or .xmbl) in the Experiments folder on the desktop. This will start the program Logger Pro3.3 and bring up the appropriate data file. If you do not have an auto-ID sensor (which is the likely case), a dialog box will 1 The theory section of this experiment was written by Jennifer LK Whalen 6 of 9 LPC Physics Centripetal Force pop up asking you to confirm the sensors being used. If you have the suggested sensor attached to the LabPro in the suggested port, click “OK”. If the “OK” button is not active, ask your instructor for help. 7. Click on ExperimentÆSet up sensorsÆLab Pro, then click on the photogate icon. Change the setting to “pulse timing.” 8. Place the index in an intermediate location and measure R, the radius of circular motion. 9. Reposition m1 so that it is directly over the index. Rotate the apparatus and reposition the counter weight so that it rotates smoothly. 10. Attach the spring to m1. 11. Rotate the shaft so that m1, while rotating, stretches the spring and lines up with the index. You will need to concentrate on keeping the weight directly over the shaft. Don’t worry if you are off slightly—but avoid systematic errors (consistently too short or too long). By repeating the experiment, you can eliminate random errors. 12. Place the photogate so that the beam will be interrupted when the hanging mass rotates at the proper radius. To avoid damaging the photogate, start with the gate too low and raise it into position once the mass is revolving at the proper radius. 13. Determine the period of rotation by timing 30 - 50 revolutions (T = t/Nrev). Repeat at least 3 times. 14. Using the set of weights, determine the force required to stretch the spring the same amount as when m1 was rotating. While you are at it, try adding and subtracting a small amount of weight (how much weight can you add or subtract so that you don't see any measurable change in the spring) to find Fmax and Fmin . These will allow you to determine your experimental uncertainty. 15. Move the index closer to the shaft (for a smaller radius) and repeat steps 8 - 14. 16. Move the index farther from the shaft, repeat steps 8 – 14. 17. Copy your data from Logger Pro to an Excel spreadsheet or Graphical Analysis to perform data analysis. NOTE: Because the pendulum timer expects its beam to be broken twice per period, your ∆t values are twice what they should be. Analysis: 1. Starting from Newton’s Second Law, and the equation for centripetal motion: 7 of 9 LPC Physics Centripetal Force F = ma, a = v2 R derive the following relationship: Fc = 4π 2 mR T2 Eq. 2 where T is the period of one revolution. 2. Using your data, and Excel or Graphical Analysis spreadsheets, determine average values of the centripetal force for each of the three radii in the experiment. For convenience’s sake, these will be referred to as the “theoretical values” of the centripetal force. 3. Determine the standard deviation for each of the values of Fc you calculated in Step 2. This will be your theoretical uncertainty in the experiment. 4. Calculate the force (m2g) necessary to stretch the spring the same amount as provided by the centripetal force. This will be referred to as the “experimental value” of the centripetal force. 5. Determine the uncertainty in the experimental value, given by (Fmax - Fmin)/2. 6. Determine the percent difference between the experimental and theoretical values in this experiment. Determine whether the percent difference between the theoretical and experimental values is less then the percent uncertainty in each case. 7. If the values are all within the total uncertainty, discuss methods of improving the experiment so that the uncertainty is reduced. If the values are not within the uncertainty, suggest possible reasons why. As in all experiments, discuss and interpret trends or anomalies suggested by the data. Results: Write at least one paragraph describing the following: • what you expected to learn about the lab (i.e. what was the reason for conducting the experiment?) • your results, and what you learned from them • Think of at least one other experiment might you perform to verify these results • Think of at least one new question or problem that could be answered with the physics you have learned in this laboratory, or be extrapolated from the ideas in this laboratory. Clean-Up: 8 of 9 LPC Physics Centripetal Force Before you can leave the classroom, you must clean up your equipment, and have your instructor sign below. How you divide clean-up duties between lab members is up to you. Clean-up involves: • Completely dismantling the experimental setup • Removing tape from anything you put tape on • Drying-off any wet equipment • Putting away equipment in proper boxes (if applicable) • Returning equipment to proper cabinets, or to the cart at the front of the room • Throwing away pieces of string, paper, and other detritus (i.e. your water bottles) • Shutting down the computer • Anything else that needs to be done to return the room to its pristine, pre lab form. I certify that the equipment used by ________________________ has been cleaned up. (student’s name) ______________________________ , _______________. (instructor’s name) (date) 9 of 9