A mass hanging at rest from a spring is acted on by two forces:
the force due to gravity (F = mg), which acts straight down towards the center of the earth, and
the force due to the spring (F = -kx0), which acts in the opposite direction to the
displacement of the spring.
If the mass is pulled downward a distance x from its equilibrium position and released, Newton’s Second Law of Motion (F = ma), can be used to write an equation of motion describing the mass’s position as a function of time.
ma = -kx
1. The acceleration, a, is equal to the second derivative of x. Collect all terms on the left side of the equal sign and rewrite the equation in terms of x and its second derivative.
2. For this type of equation, it is common to divide each term by m, and to define 02 = k/m,
where is called the angular frequency. Rewrite the above equation using 0.
This is a second-order linear differential equation. One approach to the solution of this type of equation is to guess the solution and then verify that the guess satisfies the differential equation.
Interactive Mathematics Lecture Demonstration
3. Show that the function x = sin(0t) and x = cos(0t) are both solutions to the differential
equation
5. Show that for arbitrary constants A and B that
x = Asin(0t) + Bcos(0t)
is also a solution to the differential equation.
What we want to do now is to use the general solution x = A sin(0t) + Bcos(0t), to solve the
initial value problem that consists of the differential equation
and the initial conditions, x(0) = x0 and .
6. For the general solution x = A sin(0t) + Bcos(0t), apply the initial conditions and solve for A
and B.
This type of motion is called simple harmonic motion.
Interactive Mathematics Lecture Demonstration
To see how well this equation models the actual motion of a mass-spring system, it is necessary to find values for the two parameters in the equation,
7. How can you measure x0?
8. How can you measure 0?
Appendix 1. Laboratory Measured Data of Mass-Spring Oscillation
Time Distance Time Distance Time Distance (sec) (m) (sec) (m) (sec) (m)
0 0.830123 0.025 0.818461 0.05 0.807656 0.075 0.797709 0.1 0.788791 0.125 0.780731 0.15 0.774214 0.175 0.769412 0.2 0.766153 0.225 0.764781 0.25 0.76461 0.275 0.766668 0.3 0.770441 0.325 0.775757 0.35 0.782446 0.375 0.790506 0.4 0.801654 0.425 0.811944 0.45 0.823092 0.475 0.834582 0.5 0.846758 0.525 0.85842 0.55 0.869568 0.575 0.880201 0.6 0.888261 0.625 0.897351 0.65 0.905583 0.675 0.912614 0.7 0.919646 0.725 0.92359 0.75 0.925648 0.775 0.925991 0.8 0.924619 0.825 0.919131 0.85 0.914158 0.875 0.907812 0.9 0.900266 0.925 0.891348 0.95 0.883974 0.975 0.873513 1 0.862536 1.025 0.851046 1.05 0.839041 1.075 0.82755 1.1 0.815888 1.125 0.803369 1.15 0.793765 1.175 0.785361 1.2 0.778158 1.225 0.772156 1.25 0.76804 1.275 0.765467 1.3 0.764781 1.325 0.765639 1.35 0.768383 1.375 0.772156 1.4 0.778158 1.425 0.785533 1.45 0.794108 1.475 0.803712 1.5 0.814002 1.525 0.825149 1.55 0.836983 1.575 0.848645 1.6 0.860478 1.625 0.871626 1.65 0.881916 1.675 0.891005 1.7 0.900095 1.725 0.907812 1.75 0.913987 1.775 0.918617 1.8 0.921189 1.825 0.922733 1.85 0.922219 1.875 0.92016 1.9 0.916387 1.925 0.910728 1.95 0.903868 1.975 0.895636 2 0.887061 2.025 0.879 2.05 0.868024 2.075 0.857048 2.1 0.845387 2.125 0.833896 2.15 0.822577 2.175 0.811772 2.2 0.801654 2.225 0.792736 2.25 0.784847 2.275 0.778502 2.3 0.773699 2.325 0.769926 2.35 0.768383 2.375 0.768383 2.4 0.770098 2.425 0.773699 2.45 0.778673 2.475 0.784847 2.5 0.792907 2.525 0.802168 2.55 0.812115 2.575 0.82292 2.6 0.833896 2.625 0.845387 2.65 0.85722 2.675 0.86871 2.7 0.879515 2.725 0.889119 2.75 0.895636 2.775 0.903868 2.8 0.910728 2.825 0.916216 2.85 0.919989 2.875 0.922219 2.9 0.922219
Interactive Mathematics Lecture Demonstration
Instructor Notes:
Learning Outcomes:
Upon completion of this module the students should be able to:
Identify a second order ordinary differential equation,
Solve a second order differential equation, and
Fit the solution to a second order differential equation to experimental data.
Equipment: mass, spring, stand to hold the spring, yardstick and a stopwatch.
1 .
2.
4.
6.
7. x0 is the initial displacement of the mass from its equilibrium position
8. 0 can be measured in two different ways:
a) Static method: Measure the spring constant and the mass, then To measure the
spring constant, apply a mass (m) to the spring and measure the displacement (x) of the spring. The spring constant is . Where m is in kilograms, x is in meters, and g = 9.8 m/s2.
b Dynamic method: Measure the period, T, of oscillation, then
The data in the Appendix can be pasted into an Excel spreadsheet so students can test there equation with the data.
Interactive Mathematics Lecture Demonstrations
