Names ________________ _______________ _______________ Section ______
SIMPLE HARMONIC MOTION
Introduction
The objective of this workshop is to measure the time­dependent displacement, velocity and acceleration of an object undergoing simple harmonic motion. An object in simple harmonic motion oscillates about its rest point xo with a position x given by x(t) = xo + A cos ωt,
where t is the time, A is the amplitude, and ω=2π/T is the angular frequency of oscillation. The period T is the time it takes for the object to undergo one complete cycle of oscillation. The frequency f is 1/T. If the object is a mass m hanging on a spring with force constant k, the angular frequency is
ω=
k
.
m
Taking the first time derivative of x(t) gives the velocity v of the object, v(t) = ­ ωA sin ωt.
and the second time derivative is its acceleration
a(t) = ­ ω2A cos ωt.
In this experiment you will use the motion sensor and the computer to record the displacement, velocity and acceleration of a mass suspended from a spring as it oscillates vertically. Procedure
1. Determine the spring constant k of the spring by measuring the spring’s extension under a load. First measure with a ruler the vertical position of the weight holder without any added weight, then measure the position again with the mass added (0.20 kg). TAPE THE MASS TO THE HOLDER. The spring constant is the ratio of the added weight ∆W to the change in height ∆x. Don’t forget to give the (SI) units!
∆W ______________.
k=
=
The added mass is:__________
∆x
From this, calculate the theoretical predicted angular frequency ωth, recalling that m is the total mass (weight holder + added mass). Give the units!
The total mass is:__________
2. Taking data.
ω=
k ______________.
=
m
Open the software program for this lab from the course folder. With the mass of 0.300 kg on the spring (200 g TAPED to 100 g weight holder), pull it down gently a few centimeters and set it oscillating. Put the motion sensor directly under the oscillating mass and move it slightly while recording until you get the best signal. Keep trying until you get a smooth curve in the three plots of displacement, velocity, and acceleration. If you do not see the curves, try "autoscale".
Find the first maximum in the distance, the next minimum, the second maximum, etc., and record the corresponding times, distances, velocities and accelerations in the table below. Do not forget units! It is helpful to use the cursor provided in the “x=” icon to locate and read the values at these positions. Note there is an intrinsic uncertainty in reading the positions because we have only twenty readings per second. The code does not calculate the acceleration well for the first few data points, so if your first maximum is close to the start of your data, skip to the second maximum.
distance
time
distance
velocity
acceleration
First max (1)
Next min (2)
Second max (3)
Next min (4)
Third max (5)
Now find the first maximum (most positive value) in the velocity, the next minimum (most negative value), etc., and record the corresponding distances and accelerations. Velocity
time
velocity
distance acceleration
First max Next min
Second max
3.
Calculate the following quantities from these data; be sure to give units. Note that because the zero of distance is not necessarily properly calibrated to be the equilibrium position, use (xmax – xmin)/2 for the amplitude A.
First cycle: T _________ f _________ ωexp __________ A __________
Second cycle: T _________ f _________ ωexp __________ A __________
Maximum velocity vmax = ___________ Maximum acceleration amax = ___________ Theory: vmax = ωthA = _____________ amax = ω2thA = ____________
4. Conclusions
Compare the angular frequency ωexp with the theoretical predicted value from page 1.
How do the measured vmax and amax compare with the theory values on the bottom of page 2? (Some difference may be attributed to calibration of the motion sensor.)
Fill in the blanks in a.­d. with one or more of the words “maximum”, “minimum”, and “zero”:
a. When the distance is maximum, the velocity is ___________.
b. When the distance is the equilibrium position, ie, halfway between max and min, the velocity is __________________________.
Remember that the kinetic energy of the mass on the spring is mv2/2, and the potential energy of the mass/spring system is kx2/2. Thus:
c. When the PE is maximum, the KE is _______________.
d. When the KE is maximum, the PE is ________________.