Solving simple harmonic motion
1. Understand the problem
Read the problem carefully and identify what is being asked and what values are given (e.g.,
amplitude, period, frequency, mass, spring constant, etc.).
2. Draw and label a reference circle
Clarify step 1, estimate a rough answer or range that the value will be within and define the
quadrant you’ll be working in.
3. Identify key SHM equations
a. Defining equation for SHM - acceleration and displacement are proportional and
opposite in direction.
𝐹 = −𝑘𝑦
𝑎 = −𝜔2 𝑦
b. Time Period – time taken for 1 complete oscillation.
1
𝑙
𝑚
𝑓
𝑔
𝑘
𝑇 = , for a pendulum 𝑇 = 2𝜋√ , for a spring 𝑇 = 2𝜋√
c. Angular Frequency- rate of change of angular displacement.
𝜔=
∆𝜃
∆𝑡
𝜔 = 2𝜋𝑓
𝜔=
2𝜋
𝑇
d. If the oscillation begins from the equilibrium, use:
e. If the oscillation begins from the maximum displacement, use:
4. Set up the equations based on given data and solve
Use the given values to plug into the SHM equations and solve. CALCULATOR IN RAD MODE.
5. Check units and ‘reasonableness’
Ensure that all units are consistent remembering that displacement, velocity and
acceleration are all linear quantities and need linear units. Check if the answer makes sense
in the context of the problem.
Problem
___m
A mass of 0.5 kg is attached to a spring with a spring
constant of 200 N/m. The mass is displaced 0.1 m from its
equilibrium position and released. Find the period of the
motion and the displacement after 0.2 seconds.
1. Understand the problem
m = ___kg, k = ____N/m, A = ____m, t = ___s, y= ?m
2.
Reference circles
Equilibrium is ________ line, phasor rotates ___clockwise.
Find T, using spring equation: 𝑇 =
Start (max displacement)
t= s
We know from this that at t=0.2s the spring will be more
than halfway (
s) through the oscillation but less than ¾
through (0.235s), must be close to
, y=0. So we
are working in the ____________________ part of the
reference circle.
3. Identify Key SHM Equations
Angular Frequency
Displacement when oscillation begins at max displacement 𝑦 =
4. Set Up the equations based on given data and solve
For time period of a spring
Angular Frequency
Displacement equation when oscillation begins at max displacement
5. Check units and ‘reasonableness’
The answer is _______ to the equilibrium, it is ______ than the amplitude which it must
be, and it is __________, meaning it is on the ____________ of the equilibrium from
where is started. Units are in ___.
Problem- solved
A mass of 0.5 kg is attached to a spring with a spring constant of 200
N/m. The mass is displaced 0.1 m from its equilibrium position and
released. Find the period of the motion and the displacement after
0.2 seconds.
0.1m
1. Understand the problem
m = 0.5 kg, k = 200 N/m, A = 0.1m, t = 0.2s, y= ?m
2.
Equilibrium drawn as a vertical line because the oscillation is
side to side.
Start (max displacement)
Using spring equation: 𝑇 = 2𝜋√
0.5
200
= 𝟎. 𝟑𝟏𝟒𝟐𝐬
We know from this that at t=0.2s the spring will be more than
halfway (1.57s) through the oscillation but less than ¾ through
(0.235s), must be close to equilibrium, y = 0. So we are working
in the bottom left quadrant of my reference circle.
t=0.2s
3. Identify Key SHM Equations
2𝜋
Angular Frequency 𝜔 = 𝑇
Displacement when oscillation begins at max displacement 𝑦 = 𝐴𝑐𝑜𝑠𝜔𝑡
4. Set Up the equations based on given data and solve
0.5
For time period of a spring 𝑇 = 2𝜋√200 = 𝟎. 𝟑𝟏𝟒𝟐𝒔
2𝜋
Angular Frequency 𝜔 = 0.3142 = 𝟐𝟎𝒓𝒂𝒅𝒔−𝟏
Displacement when oscillation begins at max displacement
𝑦 = 0.1 cos(20𝑥0.2) = −𝟎. 𝟎𝟔𝟓𝟑𝒎
5. Check units and ‘reasonableness’
The answer is quite close to the equilibrium, 0m, it is less than the amplitude which it must be, and it is
negative, meaning it is on the other side of the equilibrium from where is started. Units are in m.