MASS ON A SPRING - GENERAL SOLUTION

System: Mass attached to a horizontal spring resting on a horizontal, frictionless surface.

Newton's Second Law

Second Order Differential Equation

Solution as Function of Time

A = Amplitude (Considered to be a positive Constant)
= Maximum distance of the mass from equilibrium position
xmax = ± A Maximum position from equilibrium
(Occurs when w t + f = ±p/2 or ±3p/2)
vmax = ± Aw Maximum velocity
(Occurs when w t + f = 0 or ±p)
amax = ± Aw2 Maximum acceleration
(Occurs when w t + f = p/2 or ±3p/2)
f = Phase Angle (SI: rad)
w t + f = Circular motion rotational angle for SHM (SI: rad/s)
Angular Frequency of Oscillation (SI: rad/s)
(±p is determined by initial conditions)
Period of one Complete Oscillation (SI: sec)


Block on a Spring
A block is connected to a horizontal spring and rests on a frictionless surface. In this simulation the amplitude, angular frequency, phase angle, and the mass can be varied. Displayed are the position, velocity, and acceleration of the block as a function of time as well as the resulting values of the period, spring constant, and mechanical energy. Block on a Spring Problem
SHM of Mass on a Vertical Spring
A mass is attached to a vertical spring and set into motion. The initial position, initial velocity, spring constant, and mass can be varied. Displayed is the position, velocity, and acceleration as function of time. Also calculated are the resulting values of angular frequency, phase angle, amplitude, and maximum velocity. The effects of gravity can observed by turning gravity on and off.

Conservation of Energy

Energy of Oscillating Bob Simulation
A bob oscillates vertically on the end of a spring. The various forms of energy are displayed: the bob's gravitational potential energy, the spring's potential energy, the bob's kinetic energy, and the total mechanical energy of the system of the bob and spring.

Relationships Table

Example for a block is pulled to the right and released from rest.

* Except for time of occurrence, the relationships between maximum and minimum of the position, velocity, acceleration, kinetic energy, and potential energy are independent of the phase angle f, i.e. how the system is set into motion. For example, the velocity is always maximum when the acceleration is zero but this moment does not always occur at a quarter of the way through the period. However, the time between the events above are equal to 1/4 of the period. For example, the time between when the acceleration is at a maximum and when it is zero is 1/4 T.