PHYS 211 Lab 8 1 Lab 8: Centripetal Acceleration Introduction: In this lab you will confirm Newton s Second Law of Motion by examining the dynamic and static force exerted on a mass by a spring. The dynamic force exerted on the mass causes acceleration, which is centripetal acceleration in this case. Theory: To confirm Newton s Second Law, you will examine a mass undergoing uniform circular motion, or motion in a circle or circular arc at a constant speed. Although the speed of an object in uniform circular motion does not vary, the object is accelerating because the velocity changes in direction. The type of acceleration associated with uniform circular motion is called centripetal acceleration, a c. The magnitude of this acceleration is a c = v 2 (1) r where v is the velocity of the object and r is the radius of the circle the object is moving in. The units of centripetal acceleration are m/s 2. During centripetal acceleration at constant speed, the object travels a distance equal to the circumference of a circle with radius r, a distance of 2πr, in time T = 2πr (2) v where T is called the period of revolution, or, simply, the period. The period is measured in units of seconds, r is measured in meters, and velocity is measured in m/s. The period, in general, is the time it takes the object to go around a closed circle once. For an object to accelerate, a force F must act on it. This is Newton s Second Law of Motion, which specifically says that the acceleration of an object is proportional to the net outside force, F net, acting on the object. The statement of this law as an equation is F net = ma (3) where m is the mass of the object in kg and a is the acceleration of the object in m/s 2. The units of force are Newtons (N) (1 N= 1 kgm/s 2 ). By this law, for an object to be accelerating under uniform circular motion, a force must be acting on it. In this lab a mass m is attached to a spring and is rotated about the end of the spring by a motor. The spring applies a force on the mass, which gives it a centripetal acceleration. For this case, Newton s Second Law can be written as F d = m v 2 (4) r where F d is the dynamic force of the spring on the mass (see Figure 1). If we can measure the period T of rotation and we know the radius of rotation r, the velocity of the mass is v = 2πr = 2πrf (5) T
PHYS 211 Lab 8 2 where f is the frequency of the rotation (f=1/t). Figure 1. Free body diagram for the dynamic force. Combining equations 4 and 5, we find the dynamic force of the spring on the mass to be F d = 4π 2 mrf 2 (6) The spring can also exert a static force on the mass. The static spring force can be determined by hanging the mass from the spring in a vertical direction. Additional mass M can be hung from the mass m until the spring extends the same distance r as it did while rotating (see Figure 2). Figure 2. Free body diagram for the static force. When this is the case, the static force of the spring on the masses is F s = (m + M)g (7) Since the spring extension is the same for both the static and dynamic case, F s =F d. Equations 6 and 7 can therefore be equated, yielding M = 4π 2 mr g f 2 m (8) If Equation 8 can be verified, then the theoretical expression for the dynamic force of the spring (Equation 4) that causes the centripetal acceleration can also be verified. In this lab you will determine the frequency of rotation f that is proportional to the dynamic force while varying the tension of the spring. For each value of spring tension you will determine the mass M required to yield an equivalent static force. By plotting M
PHYS 211 Lab 8 3 vs. f2 and verifying that the slope of that line is equivalent to 4π2mr/g you will verify the theory yielding Equation 8. Apparatus: The apparatus consists of a metal frame with a cylindrical mass m attached to a coil spring and mounted inside it (Figure 3). The frame assembly is placed into a motor driven, variable speed rotator. The tension in the spring is adjusted by turning a threaded collar to which the spring is fastened. While at rest, the cylinder is held by the spring against a stop. When the apparatus is rotated about a vertical axis, the mass moves outward producing an extension of the spring. A pointer is loosely pivoted such that when the cylinder presses against it its tip moves upward through a range of a few mm. In operation, the speed is adjusted until the pointer is opposite its index. Since the index is practically on the axis of rotation, the position of the pointer can be clearly seen while the apparatus is rotating. Figure 3. Frame assembly of centripetal acceleration apparatus. Safety: o This apparatus rotates a metal frame at speeds that could cause injury if it were to contact your body. Be aware of where you place your hands while operating the apparatus when it is rotating at speed. o Loose clothing or long hair could become tangled in the rotating frame. Roll up your sleeves or tie back your hair while operating or working near the apparatus. o Make sure you know were the on/off switch of the apparatus is located so you can turn it off quickly if you need to.
PHYS 211 Lab 8 4 There are a few things to note about the operation of the apparatus: A revolution counter is attached to the frame of the rotator by means of a steel spring that normally holds the counter disengaged from the rotating spindle. By pressing with a finger on the end of the spring, the counter gear is engaged with an identical gear on the spindle. Some may find depressing the spring with their thumb to be more comfortable. The speed of rotation of the spindle is controlled by adjusting the point of contact of the friction disk past the center of the driving disk on the variable speed rotator. The direction of rotation is reversed by moving the friction disk past the center of the driving disk. A few minutes of operating the apparatus will teach you how it works. However, adjusting the speed so that the needle sits exactly opposite the index will take some practice. Be patient. Part I: Determine the frequency of rotation Procedure: Adjust the spring tension to a minimum by means of the threaded collar attached to the spring. 1. Record the value on the scale corresponding to this position of the threaded collar on the table provided. Make sure the centripetal force apparatus is mounted securely on the rotator spindle and that the axis of rotation is vertical. Set the friction disk so that it is near the center of the driving disk. Turn the rotator on. Adjust the direction so that the frame is rotating clockwise. Adjust the speed control until the pointer is just opposite the index. Practice regulating the speed until you are able to keep the pointer moving about the index with as little upward and downward oscillation as possible. 2. Record the initial reading on the counter, N i, in the table provided. Engage the counter gear for 30 s by pressing down the tab next to the counter. You can use the second hand on the wall clock to measure the time interval. You might have to continue to adjust the speed of the rotator while taking data, as the timing gear may slow down the apparatus. 3. Record the final reading on the counter, N f,, in the table provided. Reverse the direction of the rotator and repeat the measurement of the number of rotations in 30 s. If the clockwise and counterclockwise values for ΔN differ by more than 5%, repeat the measurements.
PHYS 211 Lab 8 5 Part II: Determine the mass M necessary to exert a static force equivalent to the dynamic force Procedure: Remove the centripetal force apparatus from the rotator and suspend it with the mass m down. Attach a mass hanger to the string attached to the mass m. Add mass until the pointer is again lined up with the index. When this is the case, the force on the spring due to the force of gravity acting on the masses m and M is equal to the dynamic force exerted when the apparatus was rotating. 4. Record the total mass M on the mass hanger. 5. Use the calipers to measure the distance r between the axis of rotation (indicated by a scribed line on the frame) and the center of the mass m. Record the measured value of r in your lab notebook. Change the tension of the spring and repeat the procedures of Parts I and II. Do this for at least five different spring tensions. Fill in the table provided as you work. Cut out the table and insert it into your lab notebook. Rotation direction Spring index N i N f ΔN Δt (s) T (s) f (s -1 ) M (kg)
PHYS 211 Lab 8 6 Part III: Plot the data and compare the experimentally and theoretical determined slopes If M is plotted versus f 2, the slope of the resulting line should be equal to 4π 2 mr/g, if Equation 8 is true. Procedure: 6. Use the computer program Logger Pro to graph M versus f 2. Input your data for M (in kg) and f 2 (in s -1 ) into two manual columns Plot your data. Make sure your axes are appropriately scaled and labeled. Turn on Point Protectors. Give your graph an appropriate title. Apply a linear fit to the data. You now have the experimentally determined slope. By double clicking the box that pops up on your graph, select Show Uncertainty to see an estimate of the error in the experimental slope (δslope) and y intercept of the linear line. Record the slope and its associated uncertainty in your lab notebook. Print your graph. Choose Landscape under Page Setup. Make sure everyone in your group gets a copy. Insert your graph into your lab notebook. 7. Use the value of r that you measured to calculate the theoretical slope 4π 2 mr/g. The mass m of the mass attached to the spring is stamped onto it. Convert this mass to kg. Use g=9.81 m/s 2. Show all your calculations in your lab notebook. Use correct units. Part IV: Error Analysis 8. Apply the appropriate rule for determining uncertainty to the theoretical expression for the slope. Calculate the uncertainty in the theoretical slope, δslope. Be sure to record your values for δm and δr and show all of your work. 9. Report your theoretical and experimental values for slope ± δslope in your lab notebook. Conclusions 10. How well do the theoretical and experimental values for the slope agree? Did you verify Newton s Second Law? Explain. 11. What should the y-intercept of the linear line fitted to your plot of M versus f 2 be equal to? Calculate the percent difference between the y-intercept given from Logger Pro and the expected value. How well do they agree? 12. What are the sources of systematic and random error in this experiment that are not accounted for in the error analysis? List at least two of each.