• Simple harmonic motion.o We will use the position and force sensors of a Smart Cart to perform the

measurement.o As the position of the cart is measured through its wheels, we place the cart on

a track, with the force sensor (the hook) attached to one end of a spring. Thespring is tapered to account for its own weight, and keep an effective springconstant the same throughout the length of the spring. The smaller end of thespring is attached to the top of the track.

o The track is placed almost vertically, with a small angle just enough for the cartto roll on it. You will need to make sure that the side of the track is vertical witha level, and the wheels are in the grooves on the track. Please see the photo onthe right. If you lean the track against the table in the lab, lower a track foot tothe table level and use C-clamps to secure it to the table.

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PHYS 1105, Module 4

Simple harmonic oscillation and 2-dimensional motion: projectile and circular

o Now turn on the power of the cart and use Capstone (ex. oscillator.cap) to connect to it. In hardwaresettings you will need to select the following sensors: position, velocity, acceleration and force. In theposition and the force sensors, with the cart hanging on the spring in equilibrium, zero the sensors sothat the measurements will be with respect to this position. Construct your measurement with position,velocity, acceleration and force, and select a data rate 50 Hz or above.

o Now start the measurement, and then pull the cart about 10 cm down and let go. The cart will oscillateon the track. Please make sure the wheels are always moving in the groove. If not, the track side is notvertical. Record 5 or 6 oscillations and stop the data taking. You should have a screen like this

o Find the legend that corresponds to position (x) and start the measurement when the cart starts theoscillation. One can read out the period T and amplitude A from the graph directly. One can also use thedata of the position to read out these two pieces of information with higher precision. Mostly due tofriction, the cart slows down and the amplitude reduces (damped). This is evident in the plot. Now findthe legends of the other three measurements, and see when they research maximum and minimum(magnitude only) and compare what you observe with the predictions from the formulas (pre_lab). Payattention to the direction of the force (acceleration as well) and see if the force is always pointing to theopposite direction of displacement. Understand this with the definition of restoring force.

o Data analysis: Calculate for v and a (and optionally f, for this you will need the mass of the cart), basedon x, and compare results with measurements (example in the file data_recording_and_analysis.xlsx).Ambitious students are encouraged to fit the data in Capstone to a damped oscillation.

Measurement starts here

1 T

4 T

A

A

• Simple harmonic motion - continued. o The spring constant is not given in the setup. Design a measurement to find it out, and more importantly,

also find out the measurement error of the value you quote as the result. Perform the measurement.

• Projectile motion. o In this study we will use the launcher with a dual beam photo-gate (dual_beam_photo_gate.cap for data

acquisition) to launch a steel ball and measure the initial (muzzle) velocity. The launcher also provides the reading of the launching angle. The protractor may have systematic errors, use the level to calibrateit. Or, you can also use the provided digital protractor to verify the launching angle.

o The first study will be to understand the two components in the motion. Set the launcher at the edge of the table to launch the ball horizontally. Measure the distance from the muzzle to the floor (you may need a plumb bob to find the vertical line from the muzzle to the floor, mark this point) to determine the free-fall distance and the time. Now launch the ball with the muzzle speed measured by the photo-gate, and the ball captured with the ball-catcher (a small cardboard box or the like). Calculate for the landing point and distance with the knowledge of the muzzle speed. Make sure that the ball will have a clear flight and not to hit anyone in its path, before you actually launch the ball without the catcher. Now record the landing point with the landing pad (a letter size white paper under a carbon paper, taped to the floor) placed at the anticipated landing location. Verify your calculation with a measurement of the landing distance using a tape measure. Repeat at least 5 times at each launching speed (for short, medium and long range) to obtain the measurement errors. Design your data recording format, and decide to use Capstone or excel spreadsheet to record the raw data. An example is provided as in the file data_recording_and_analysis.xlsx. You will need to provide a copy of the raw data in your lab report. For different launching speeds, do you see the effects of air resistance showing up in certain launching distances? Discuss this with the TA and in your lab report.

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Simple harmonic oscillation and 2-dimensional motion: projectile and circular

Use level to calibrate the protractor. When it’s leveled, the protractor should read 0. In this particular case the calibration is -3 degrees. This willbe needed in the angled launch.

Make sure that the launcher is leveled. Use the plumb bob and the tape measure to find the muzzle height above the floor and its vertical point on the floor. Mark this position on a paper taped on the floor. The horizontal launching distance will be measure with respect to it.

Set up the launcher as in the photos. Use the screw in the middle.

• Projectile motion - continued. o The second study will be to find the launching angle for the farthest landing distance. You only need to

do this with the launcher set for “medium range”. You clamp the launcher to the table edge. The launcher faces inwards to the table and the muzzle is set to be flush with the table top (see the photo below). With the knowledge of the “medium range” muzzle speed from the previous measurements, calculate for the landing distance at 45 degree launching angle, and make sure that the ball will land on the top of the table. Now launch the ball from 35 degrees to 55 degrees, with 5 degree steps, and measure the landing distance with the landing pad. Each time also measure the muzzle speed. Again perform 5 launches at each angle to estimate measurement errors. Design your data recording format and record data.

o Analyze your measurement data and find out the angle with the longest landing distance. You may use interpolation or a fitted function to obtain the angle for the longest landing distance if that’s not one of your measured data points. Compare the result with calculation.

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Simple harmonic oscillation and 2-dimensional motion: projectile and circular

Mount the launcher to the edge of the table, facing inwards. Use the screw at the muzzle so that when you change the angle, the muzzle height will not change. The lower level of the muzzle is set to be almost flush with the table top.

The lower level of the muzzle is almost flush with the table top.

Apply the calibration when setting the angles. For example, 45 degree should read 42 in this particular case.

• Uniform circular motion. o What we want to verify in this measurement is the formula that links the centripetal force with the

rotating velocities: 𝐹 = 𝑚𝑟𝜔! = 𝑚 "!

#. That is, F changes linearly with 𝜔! or 𝑣! once m and r are set.

o The setup is based on two instruments: the force sensor that also measures its rotating angular velocity and the dual-beam photo-gate that measures speed. The rotor turned by a motor with a reduction gear provides the rotation to the steel ball in a protection tube. The steel ball is connected to the force sensor through a string. The force sensor is placed on the rotor in such a way that the force sensor itself is at the center to avoid any contribution to the force measurement due to the mass of the sensor and the hook. A DC power supply drives the motor. You change the voltage (please do not go over 12 V) which changes the rotating speed. The angular velocity is measured by the force sensor (that is the z component of the angular velocity). The end of the rotor passes through the photogate that measures the linear velocity (or speed) through the two beams. Weigh the steel ball (the mass m) and attach it to the force sensor. Measure the distance of the center of the ball to the center of the rotor (the radius r). Zero the force sensor before you turn on the power to the motor. Set the rotor in motion and you should be able to measure the centripetal force to the ball by the string and the angular and linear velocities, using Capstone. You are encouraged to design the measurement and record the data yourself. You may also choose to use the file UCM.cap as a starting point. In this file we plot the force versus 𝜔!or 𝑣! to verify the linear relationship. You can ramp up and down the voltage (again please do not go over 12 V) to obtain a full measurement. You can shorten the string length by using a loop-back to halve its length and repeat the measurement one more time. You can also make your own string and do the measurement at different r values. Record data and compare the measurement with your calculation. Discuss possible measurement errors in your report.

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Simple harmonic oscillation and 2-dimensional motion: projectile and circular

force sensor

Photo gate

Force v.s. velocity squared, and a linear fit. In this measurement the distance of the ball to the center is 13.5 cm. The linear velocity measured by the photo gate is at a distance 17.5 cm from the center. This corresponds to a linear velocity of the ball at 𝑣$ = ⁄13.5 17.5 𝑣%&.Using the linear fit in Capstone, one has 𝜔! = 100𝐹 = '

(#"𝐹, '

(#"= '

).)++×).'-.= 112

in the SI units. Given the error on 𝑟$, this is very close. The other fit generates:

𝐹 = 𝑚 ""!

#"= #"

('-..'/..

𝑣%&!

and this leads to 𝑣%&! = #"(

'/..'-..

!𝐹 = 2.87𝐹. Converting all units to SI, we

find out #"(

'/..'-..

!= ).'-.

).)++'/..'-..

!= 3.4 again given the errors on the distance measurements, this is

close to be 2.87. Please perform the same analysis on your data.