Simple Harmonic Motion
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1 [International Campus Lab] Objective Investigate simple harmonic motion using an oscillating spring and a simple pendulum. Theory Reference Young & Freedman, University Physics (4 th ed.), Pearson, (p.459~466) 4.4 Applications of SHM Vertical SHM (p.470~47) 4.5 The Simple Pendulum (p.474~475) Periodic motion or oscillation refers to any movement of an object that is repeated in a given length of time. Fig. shows one of the simplest systems that can have periodic motion. Whenever the body is displaced from its equilibrium position, the spring force tends to restore it to the equilibrium position. We call a force with this character a restoring force. Oscillation can occur only when there is a restoring force tending to return the system to equilibrium. If the spring is an ideal one that obeys Hooke s law, the restoring force FF xx is directly proportional to the displacement from equilibrium xx. The constant of proportionality between FF xx and xx is the spring constant kk. Then, FF xx = kkkk () Fig. Model for periodic motion. When the body is displaced from its equilibrium position at xx = 0, the spring exerts a restoring force back toward the equilibrium position. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 2983, KOREA ( ) Page / 9
2 When the restoring force is directly proportional to the displacement from equilibrium, as given by Eq. (), the oscillation is called simple harmonic motion, abbreviated SHM. If we use a hanging spring and a body is set in vertical motion, it also oscillates in SHM with the same angular frequency as though it were horizontal. In general, the restoring force depends on displacement in a more complicated way than in Eq. (). But in many systems the restoring force is approximately proportional to displacement if the displacement is sufficiently small. Thus if the amplitude is small enough, we can use SHM as an approximate model for many different periodic motions. By substituting the Newton s second law FF xx = mmaa xx into Eq. () and rearranging the terms, the acceleration aa xx of a body in SHM is given by aa xx = dd2 xx dddd 2 = kk mm xx (2) And now we can express the displacement xx of the oscillating body as a function of time tt. dd 2 xx dddd 2 + kk mm xx = 0 (3) Suppose we hang a spring with force constant kk (Fig. 2) and suspend from it a body with mass mm. The body hangs at rest, in equilibrium. In this position the spring is stretched an amount Δll just great enough that the spring s upward vertical force kkδll on the body balances its weight mmmm, so kkδll = mmmm. Take xx = 0 to be this equilibrium position and take the positive xx-direction to be upward. When the body is a distance xx above its equilibrium position, the extension of the spring is Δll xx. The upward force it exerts on the body is then kk(δll xx), and the net xx-component of force on the body is FF net = kk(δll xx) + ( mmmm) = kkkk. So vertical SHM doesn t differ in any essential way from horizontal SHM. The only real change is that the equilibrium position xx = 0 no longer corresponds to the point at which the spring is unstretched. If we define angular frequency ωω as Eq. (4), the solution of Eq. (3) becomes Eq. (5) or (6). ωω = kk mm (4) xx = aa cos ωωωω + bb sin ωωωω (5) xx = AA cos(ωωωω + φφ) (6) The corresponding frequency ff and period TT relationships are ff = ωω 2ππ = 2ππ kk mm (7) TT = ff = 2ππ ωω = 2ππ mm kk (8) Fig. 2 A body attached to a hanging spring 85 Songdogwahak-ro, Yeonsu-gu, Incheon 2983, KOREA ( ) Page 2 / 9
3 A simple pendulum is an idealized model consisting of a point mass suspended by a massless, unstretchable string. The corresponding frequency and period relationships are When the point mass is pulled to one side of its straightdown equilibrium position and released, it oscillates about the equilibrium position. ff = ωω 2ππ = 2ππ gg LL TT = ff = 2ππ ωω = 2ππ LL gg (2) (3) The path of the point mass is not a straight line but the arc of a circle with radius LL equal to the length of the string (Fig. 3). We use as our coordinate the distance xx measured along the arc. If the motion is simple harmonic, the restoring force must be directly proportional to xx or θθ = xx LL. Note that these expressions do not involve the mass of the particle. For small oscillations, the period of a pendulum for a given value of gg is determined entirely by its length LL. A long pendulum has a longer period than a shorter one. The restoring force is provided by gravity; the tension TT merely acts to make the point mass move in an arc. We represent the forces on the mass in terms of tangential and radial components. The restoring force FF θθ is the tangential component of the net force: FF θθ = mmmm sin θθ (9) We emphasize again that the motion of a pendulum is only approximately simple harmonic. When the amplitude is not small, the departures from simple harmonic motion can be substantial. The period can be expressed by an infinite series; when the maximum angular displacement is Θ, the period is given by The restoring force FF θθ is proportional not to θθ but to sin θθ, so the motion is not simple harmonic. However, if the angle θθ is small, sin θθ is very nearly equal to θθ in radians. With this approximation, Eq. (9) becomes TT = 2ππ LL 2 + Θ gg 2 2 sin Θ sin4 + (4) 2 When Θ = 5, the true period is longer than that given by approximate by Eq. (3) by less than 0.5%. FF θθ = mmmmmm = mmmm xx LL or FF θθ = mmmm xx (0) LL The restoring force is then proportional to the coordinate for small displacements, and the force constant is kk = mmmm LL. From Eq. (4) the angular frequency ωω of a simple pendulum with small amplitude is ωω = kk mm = mmmm LL mm = gg LL () Fig. 3 An idealized simple pendulum 85 Songdogwahak-ro, Yeonsu-gu, Incheon 2983, KOREA ( ) Page 3 / 9
4 Equipment. List Item(s) Qty. Description PC / Software Video Analysis: SG PRO Records, displays and analyzes videos. Camera Tripod Screen Feeds or streams its image in real time to a computer. Supports a camera. PVC foam board, white, mm Meter Stick Measures the length of pendulums. Spring Exerts a restoring force back toward an equilibrium position. Weight Hanger Designed to hang several holed weights. - Polycarbonate base with steel post - Mass: approx. 5g Weights (Disk) set Holed weights - Mass: approx. 5g, 0g, 20g( 2), 50g Weight (Cylinder) Weight with yellow band Weight (Ball) Green plastic ball Thread Scissors Suspends a ball weight to form a simple pendulum. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 2983, KOREA ( ) Page 4 / 9
5 Item(s) Qty. Description Three-Finger Clamp Holds pendulums. Multi-clamp Provides stable support for experiment set-ups. A-shaped Base Support Rod (00mm) Provide stable support for experiment set-ups. Electronic Balance Measures mass with a precision to 0.0g. Setup Setup. Equipment setup Weights Details: Disk Cylinder Ball Use Expt. Spring Constant Expt. 2 Motion of Mass Expt. 3 Simple Pendulum Shape Flat Disk Center Hole with Hook Yellow Band (for auto-track) with Hook Green Plastic (for auto-track) Image Setup2. Software Setup (SG PRO) If you are new to SG PRO software, see Motion of a Rigid- Body lab manual. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 2983, KOREA ( ) Page 5 / 9
6 Procedure Experiment. Spring Constant (4) Measure the change in length of the spring. () Measure the mass of the weight hanger and the weights. Use the electronic balance to measure mass. - Weight Hanger: approx. 5g - Weights: approx. 5g, 0g, 20g( 2), 50g each Record the position of the reference point on the meter stick, when different amounts of mass are added onto the weight hanger. Do not forget to include the mass of the weight hanger (5g) in your calculation of the total mass. Determine the change in length of the spring for mass varying from 40 to 80g in steps of 5g. (2) Set up equipment. Suspend the spring so that it hangs vertically. Put the weight hanger on the end of the spring. mm (kg) FF = mmgg (N) xx (m) Note (3) Specify your reference point of measurement. For your reference, choose any measuring point such as the bottom end of the spring. When analyzing your data, you have to take into account the initial tension in the spring. To ensure consistent rest lengths, most spring manufacturers design extension springs with an initial tension, which keeps the coils pressed tightly together. Hooke s law may not work for small applied forces, as you must first overcome any initial tension before you see any apparent change in length. (5) Find the spring constant kk. Find the slope of FF-xx graph using the method of least squares (see the appendix of Free Fall lab manual). The spring constant is equal to the slope of the FF-xx graph. (6) Repeat measurement. Repeat steps (4) to (5) more than 3 times. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 2983, KOREA ( ) Page 6 / 9
7 Experiment 2. Motion of a Mass on a Spring (6) Analyze your result. () Measure the mass of the cylinder weight. Use the electronic balance to measure mass. (Ignore the marked value on the weight.) Select [T-Y] (time vs. yy-axis) graph. 2 Change [ 표시방법 ](display type) to [ 선 ](line). (2) Set up equipment. Set the meter stick aside. 2 Place the camera and the screen. 3 Suspend the spring and attach the weight at the end of the spring. 3 Right-click on the graph and select [ 십자선추가 ](Show Crosshairs) from the list, and then click anywhere to show crosshairs. Drag the crosshairs to read off the coordinates of the graph. (3) Run SG PRO software. See the Motion of a Rigid-Body lab manual. You don t have to calibrate the video scale because you will measure an elapsed time, not a distance or a length. (4) Let the weight oscillate vertically. When displacing the weight, do not stretch the spring more than about 2cm from its equilibrium position. (5) Record a video. Save the video clip including about 5 oscillations. 4 Read off the coordinates of every peaks and calculate the period TT of the motion. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 2983, KOREA ( ) Page 7 / 9
8 (7) Repeat the experiment. (3) Let the pendulum swing. Repeat steps (4) to (6) more than 3 times. Displace the pendulum about 5 from its equilibrium position and let it swing. (8) Analyze the result. (4) Record a video and analyze the result. Using the spring constant kk (expt.) and the mass mm of the weight (step()), calculate Eq. (8). Using [T-X] (time vs. xx-axis) graph, find the period of oscillations of the simple pendulum and verify Eq. (3). TT = 2ππ mm kk (8) TT = 2ππ LL gg (3) (5) Vary the length LL of the pendulum and repeat the experiment. Experiment 3. Simple Pendulum Repeat the procedure of expt. 2 using a simple pendulum. () Set up equipment. (6) (Optional) Find the period of the simple pendulum when the amplitude is not small. When the maximum angular displacement Θ is not small, verify the period TT is given by Use a piece of thread and the green ball weight to make a simple pendulum. TT = 2ππ LL 2 + Θ gg 2 2 sin Θ sin4 + (4) 2 (2) Measure the length LL of the pendulum. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 2983, KOREA ( ) Page 8 / 9
9 Result & Discussion Your TA will inform you of the guidelines for writing the laboratory report during the lecture. End of LAB Checklist Please put your equipment in order as shown below. Delete your data files and empty the trash can from the lab computer. Turn off the Computer and the Interface. Leave the White Screens together at the front of the laboratory. Place the Camera and Tripod assembly on any safe place. Take care not to permanently deform the Spring. (Never stretch it beyond the elastic limit.) 85 Songdogwahak-ro, Yeonsu-gu, Incheon 2983, KOREA ( ) Page 9 / 9
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