Oscillatory Motion PHYS 01 Oscillatory Motion 1
his experiment has four parts: Oscillatory Motion 1. Determine g using a photogate and picket fence. Determine g using a simple pendulum ( different bobs) 3. Determine spring constant k for an inertial balance 4. Study the pendulum motion of your arms and legs as you walk Photogate Works like a V or DVD remote control. he computer records the times when the gate is blocked and unblocked. Picket Fence Used to measure acceleration with one photogate. Dx Dx Dx Dx Dx Dx Dx Dx PHYS 01 Oscillatory Motion
Acceleration of a Free Falling Body 1,, 3, 4, 5, 6, 7 Photogate a n 1 v n1 n1 vn n PHYS 01 Oscillatory Motion 3
he Simple Pendulum We can determine the motion of the simple pendulum by looking at the torque acting on the center of mass. q bob mg sinq mg F mg Fx he lever arm: x = F mgsinq mg sinq mg sinq mgsinq PHYS 01 Oscillatory Motion 4
he Small Angle Approximation If an angle is measured in radians and it is very small then we can use the small angle approximation: sinq q You can try this on your calculator. Set the mode to radians (rad) and take the sine of 0.1. he answer should be 0.0998, which is very close to 0.1. Now we can write the torque as: Difference 1% for q 14 0.4 rad Note that for Simple Harmonic Motion: mg sinq mgq Displacement q A mgq mg cost mg Acost Acost PHYS 01 Oscillatory Motion 5
We also know that torque is equal to: I Where I is the moment of inertia and is the angular acceleration. For Simple Harmonic Motion: a Acos t a Acos t Acost I I Acost I Acost PHYS 01 Oscillatory Motion 6
We can now put our two expressions for the torque together: I I mg Acost Acost mgacost AND Acost mg I mg I Solve for We will measure the period, which is the length of time for one oscillation. mg f I I mg PHYS 01 Oscillatory Motion 7
I mg his is the period for a Physical Pendulum. We can use this to determine the period for a simple pendulum by noting that: I m I m mg mg g Simple Pendulum g Note that this does not depend on the mass, so it should not matter if you use a large bob or a small bob. PHYS 01 Oscillatory Motion 8
g If you take a simple pendulum to the Moon would its period increase, decrease or stay the same? Assume the length stays the same. a) Increase b) Decrease c) Stay the Same What would be the period of a simple pendulum on the International Space Station? a) he same as on Earth. b) Zero c) Infinite PHYS 01 Oscillatory Motion 9
g Square both sides of the equation: 4 g Make a plot of Slope = 4 g versus Intercept = 0 PHYS 01 Oscillatory Motion 10
Basic Procedure Adjust the position of the pendulum bob so that it blocks the photogate beam. Measure the length from the support to the center of mass. Pull the bob back by less than 0 degrees and release. Measure at least 10 total oscillation periods. Change the length of the string and repeat at least 5 times. Repeat all the measurements with a different size bob. PHYS 01 Oscillatory Motion 11
he Inertial Balance Spring x Mass m Hooke s aw: Support Spring ray F kx F kx m tray a Acos t tray m m cos tray A t k Acost m m k tray m m a kx x Acost PHYS 01 Oscillatory Motion 1
As before, we will measure the period, which is the length of time for one oscillation. k f m m Square both sides of the equation: tray Inertial Balance 4 m m tray k m m tray k Slope = 4 k Make a plot of versus m Intercept = 4 k m tray PHYS 01 Oscillatory Motion 13
Basic Procedure Adjust the position of the inertial balance so that it blocks the photogate beam. Place some mass on the tray and clamp it in place. Push the tray slightly in the horizontal direction. Measure at least 10 total oscillation periods. Repeat the measurement for all combinations of masses. Remember to measure the period with the empty tray! Measure the period with the unknown mass. Compare the unknown mass values from the inertial balance and scale. PHYS 01 Oscillatory Motion 14
m m tray k If you take an inertial balance to the Moon would its period increase, decrease or stay the same? Assume the mass and spring constant stay the same. a) Increase b) Decrease c) Stay the Same What would be the period of an inertial balance on the International Space Station? a) he same as on Earth. b) Zero c) Infinite PHYS 01 Oscillatory Motion 15
Applications of the Inertial Balance NASA uses an inertial balance to measure the mass of an astronaut on the International Space Station. With a very, very small version of the inertial balance it is possible to measure the mass of a bacterium or virus. 4mm http://www.news.cornell.edu/releases/april04/attograms.ws.html http://www.news.cornell.edu/releases/nov04/virus.mass.ws.html http://www.sciencedaily.com/releases/01/08/108614358.htm PHYS 01 Oscillatory Motion 16
he Pendulum Motion of Walking When we walk, our arms and legs swing like a pendulum. Since the mass is evenly distributed along the length an arm or leg must be treated as a physical pendulum. I mg he moment of inertia I may be parameterized like so: I m Where is a pure number that depends on the distribution of mass relative to the axis of rotation. Putting this in the equation for the period gives: I m mg mg g PHYS 01 Oscillatory Motion 17
5 g Solving for gives: g You will use this equation to determine for your arms and legs by measuring the period of oscillation as you walk. You will measure the period of oscillation for your legs and arms by measuring the time it takes for you to make a fixed number of steps. Dividing the time by the number of oscillations will give you an average value for the period. You will take 11 steps which is 5 complete oscillations for one leg. You can assume that all your arms and legs have the same period. 4.5 3.5.5 1.5 0.5 10 8 6 4 11 9 7 5 3 1 5 4 3 1 PHYS 01 Oscillatory Motion 18
Basic Procedure 1. Measure the length of your leg from your hip joint to the ground.. Measure the length of your arm from your shoulder joint to your fingers. 3. Measure the time it takes for you to walk 11 steps. ake a few steps before starting the timer to get into your stride. 4. Divide the time by 5 to get the average period. 5. Calculate for your arms and legs. PHYS 01 Oscillatory Motion 19
PHYS 01 Oscillatory Motion 0