Question

Vibrational Motion

Introduction

If an object is following Hooke’s Law, then

F_net = -kx = ma

Since acceleration is the second derivative of position with respect to time, the relationship can be written as the differential equation:

kx = m δ2xδt2/{"version":"1.1","math":"<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>x</mi><mo>&#160;</mo><mo>=</mo><mo>&#160;</mo><mi>m</mi><mo>&#160;</mo><mfrac bevelled="true"><mrow><msup><mi>&#948;</mi><mn>2</mn></msup><mi>x</mi></mrow><mrow><mi>&#948;</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></math>"}

Methods for solving differential equations are beyond the scope of this course; in fact, a class in differential equations is usually a requirement for a degree in engineering or physics. However, the solution to this particular differential equation is quite well known:

x(t) = A cos (ωt + δ)

where A is the amplitude of vibration, δ is the phase shift angle and ω is the angular frequency, measured in rad/sec. The amplitude and phase shift angle are arbitrary constants; the angular frequency is determined by the conditions of the system. For a mass vibrating horizontally or vertically on a spring, the relationship is

ω = km−−√{"version":"1.1","math":"<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#969;</mi><mo>&#160;</mo><mo>=</mo><mo>&#160;</mo><msqrt><mfrac><mi>k</mi><mi>m</mi></mfrac></msqrt></math>"}

where k is the spring constant and m is the mass of the vibrating object.

The purpose of this investigation is to verify these and other relationships for a vibrating mass. At many institutions, it is possible to measure position, velocity and acceleration as a function time using appropriate hardware connected to a computer. Since that is not possible for an at-home experiment, real data has been collected as saved as a movie that is ready to be analyzed in Logger Pro.

Materials and Equipment

• Logger Pro software
• The vibrational motion.cmbl file

Data Collection

Data for a mass vibrating on a spring has already been collected. Download and open the file vibration.cmbl in a running copy of Logger Pro. The file includes a movie of a 100 gram mass hanging vertically from a spring. One end of the spring is attached to a force probe, while a motion detector is located underneath the mass. Both the force probe and the motion detector were set so that the net force and the x-position were zero when the mass was in equilibrium and at rest.

At the start of the video clip, the mass is displaced from equilibrium and data collection begins. Position, velocity, acceleration and force are recorded for five seconds. The file begins with plots of force as a function of time and position as a function of time.

Data Analysis

NOTE: On the Report Sheet, a measured value has a line after it but a calculated value has just a space. For all calculated values, write the appropriate equation first and then put in the numerical values before reporting a final number with units. Failure to do so can result in a lower score for this activity.

BASIC PARAMETERS

Use the Examine button (“x =” for a Windows system) to identify the maximum amplitude of vibration from the position vs. time plot and record it on the Report Sheet. Find the period of vibration by determining the time it takes the mass to complete one vibration. Do this by measuring the time it takes to complete a number of vibrations and dividing by that number.

Calculate the counting frequency, f, and the natural frequency, ω, for the motion.

FORCE CONSTANT AND NATURAL FREQUENCY

Open the Graph Options for the force vs. time plot. Un-check the Connect Points option. Plot position on the x-axis and force on the y-axis. The graph will demonstrate that the system is following Hooke’s Law. Use the linear fit button (‘m=’ on Windows systems) to find the force constant from the slope of the data. Record the value on the Report Sheet.

Calculate the natural frequency from the constant and the mass of the hanging weight, then express the result in terms of counting frequency and period. Compare the theoretical results to the experimental values found in the previous section.

Velocity Graph, v(t)

Change the variables on the top graph from force vs. position to velocity vs. time and check the Connect Points from the Graph Options menu. Resize the graph so that the two graphs have their time axes aligned, if that is not the case.

Since velocity is the derivative of position with respect to time, and

x(t) = A cos (ωt + δ)

then

v(t) = -Aω sin (ωt + δ)

Record the maximum velocity from the plot of velocity vs. time and compare it to the theoretical value, v_max = Aω.

Demonstrate that the functional form of velocity is correct by locating four consecutive data points from the curves. Begin with a time when the position was at a maximum positive value; followed by the time when the amplitude was zero; then when the amplitude was a maximum negative value; and finally when the amplitude was zero again. Record time, position and velocity in the data table on the Report Sheet.

Acceleration Graph, a(t)

Change the variable on the top graph from velocity on the y-axis to acceleration.

Since acceleration is the second derivative of position with respect to time, and

x(t) = A cos (ωt + δ)

then

a(t) = -Aω² cos (ωt + δ)

Note that the two curves appear to be mirror images of each other: when the position is maximum positive the acceleration is maximum negative, and vice versa. Also, both curves seem to cross the horizontal axis at the same times.

Record the maximum acceleration from the plot of acceleration vs. time and compare it to the theoretical value, a_max = Aω².

Demonstrate that the functional form of acceleration is correct by adding the acceleration of the system at the same times as position and velocity in the previous data table.

Force Graph, F(t)

Change the variable on the lower graph from position on the y-axis to force. Note that the force and acceleration graphs appear to be in phase.

Demonstrate Newton’s second law, F = ma, from the data by changing the x-axis on the force plot from time to acceleration. Un-check the Connect Points option again. Use the linear fit button to find the average value of the slope from some of the data and compare it to the mass of the system, 0.100 kg.

Questions and Conclusions

• How would the graphs and calculated values be different if a 200 gram mass was used instead of a 100 gram mass? Be specific.

Answer 1

for the given question
if m' = 200 gram was used instead of m = 100 grams
then
m' = 2m

now
the equation of motion becomes
m'dv/dt + kx = 0
hence
w' = sqrt(k/m') = w/sqrt(2)
so

the corresponding chang ein the grapsh is as under
x(t) = A*cos(w'*t + del)
v(t) = x'(t) = -Aw'*sin(w't + del)
a(t) = x"(t) = -Aw'^2*cos(w't + del)

the diffenence in position time graph will be that amplitude will be the same for sma einitial displacment and initial speed
but the frequency of osscilation will be lower so time period will be higher
hence the graph will be more spread out

for velocity time graph
the amplitude of the new system reduced over the old one by factor of sqrt(2)
and the graph is spread out

for acceleration time graph
the amplitude is half the amplitude of the old system ( as mass is double) and the graph is more spread out