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Simple Harmonic Motion

Simple harmonic motion (SHM) is defined as a repetitive back and forth motion of a mass on each side of an equilibrium position. The motion occurs between maximum displacements at both sides of the equilibrium position. The time taken to reach the same maximum displacement point is always the same. The completion of this motion on both sides is called a full cycle. The time it takes…

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Simple Harmonic Motion

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Jetzt kostenlos anmeldenSimple harmonic motion (SHM) is defined as a repetitive back and forth motion of a mass on each side of an equilibrium position. The motion occurs between maximum displacements at both sides of the equilibrium position.

- The time taken to reach the same maximum displacement point is always the same.
- The completion of this motion on both sides is called a
**full cycle**. - The time it takes for a full cycle to pass is referred to as
**period****T**. - Hence, simple harmonic motion is assumed to be a
**periodic oscillation.**

During SHM, a force known as the restoring force is created due to the acceleration of the body that results in its oscillation. This force is proportional to the displacement but has the opposite direction, causing the object to return to the equilibrium position as seen here**.**

** **

A pendulum, Georgia Panagi - Vaia Originals

An object can be assumed to oscillate in a simple harmonic motion if the following conditions are met:

Oscillations are periodic. This means that the object returns to its initial position at the same time interval for each cycle.

The acceleration of the object oscillating in a simple harmonic motion is proportional to its displacement but has an opposite direction. (Check out

**Periodic Motion**for more information.)

Here are a few examples of harmonic periodic motion.

A rocking chair that is moving back and forth as it returns to its initial position at equal time intervals.

A mass on a spring as it oscillates around the equilibrium position at equal time intervals.

A spring that is oscillating longitudinally at the same time intervals.

If a mass is attached to a spring and then displaced from its initial resting position, it will oscillate about the initial position in simple harmonic motion.

Hooke's law states that the restoring force that is required to either extend or compress the spring from its initial resting position by a distance *x*, is proportional to the spring constant k, which is a characteristic of the spring's stiffness as shown below where F is the force, k is the constant, and x is the displacement. The negative sign on the formula below indicates that the force has a negative direction from the displacement. The period of an oscillating spring can also be found using the equation below where *T *is period, and *m *is the mass of the spring.

\[\text{Hooke's Law: } F[N] = -k[N/m] \cdot [m] \qquad \text{Newton's second law: } F[N] = m[kg] A[m/s^2] \qquad T [s] = 2 \pi \sqrt{\frac{m}{k}}\]

Hooke's law has the same form as Newton's second law, where mass is the reciprocal of the spring's stiffness and acceleration is the reciprocal of the negative displacement. Hence, the acceleration in simple harmonic motion is proportional to the displacement and has the opposite direction (see below; where x is the distance of a mass oscillating from its equilibrium position).

There are various equations used to describe a mass performing simple harmonic motion.

The time period *T*, of an object performing simple harmonic motion, is the time it takes for a system to go through one full oscillation and return to its equilibrium position. One full oscillation is assumed to be completed when an object has moved from its initial position, reached the two maximum displacement points, and then returned to its initial position.

The time period can be found from the equation below, where ω is the rate of change of angular displacement with respect to time, *T *is the period, and *f *is the number of full oscillations completed in one second.

\[F [Hz] = \frac{1}{T} [s]\omega [rad/s] = \frac{2 \pi}{T[s]} = 2 \pi \cdot f\]

The maximum acceleration, *a*, of an object oscillating in simple harmonic motion is proportional to the displacement, *x, *and the angular frequency *⍵*. The formula indicates that the acceleration has an opposite direction from the displacement indicated from the minus sign. It also shows that the acceleration reaches its maximum when the displacement is at the maximum amplitude, which is the furthest point from equilibrium.

\[\alpha_{max} [m/s^2] = -\omega ^2 [rad/s] \cdot [m]\]

The given equation is described below**, **where an acceleration vs position graph illustrates that acceleration is a function of displacement. The slope of the given graph is equal to the negative squared angular frequency as shown in the equation below. The maximum and minimum displacement are therefore + x_{0} and -x_{0}, as respectively shown.

Acceleration vs displacement graph, Panagi - Vaia Originals

\[\text{slope} = \frac{a}{x} = -\omega^2[rad/s]\]

The position of an object in harmonic motion can be found using the equation below if the angular frequency and amplitude at a given time are known.

\[x(t) = x_0 \sin(\omega t)\]

This equation can be used when the object is oscillating from the initial equilibrium position. A sine graph can be used to describe this motion as shown in the figure below, which illustrates the example of a pendulum starting from the equilibrium position.

Pendulum example and Sine graph, Panagi - Vaia Originals

If an object is oscillating from its maximum displacement position where the amplitude is equal to either -x_{0} or x_{0}, then the equation below can be used.

height="25" id="2135825" \(x(t) = x_0 \cos(\omega t)\)

An illustration of a pendulum example starting to oscillate at its maximum amplitude position can be described by a cosine graph and equation as shown below.

These two graphs represent the same motion but different starting positions.

Pendulum example and Cosine graph, Georgia Panagi - Vaia Originals

The speed of an object oscillating in simple harmonic motion at any given time can be found using the equation below where *Vo *is the maximum velocity, *t *is time, and *ω* is the angular frequency.

\(V(t) = V_{max} \cos (\omega t) \qquad V_{max} = \omega \cdot x_0\)

This equation can also be derived from the position equation by deriving in terms of time; remember, velocity is the derivative of position over time. Another equation is used to describe the behaviour of speed with respect to the displacement and frequency of the harmonic oscillator shown below, where *Xo *is the amplitude and *X *is the displacement.

\(V = \pm \omega \sqrt{x_0^2 - x^2}\)

The acceleration of an object oscillating in simple harmonic motion at any given time can be found using the equation below, where *a _{max} *is the maximum acceleration,

\(a (t) = -a_{max} \cdot \cos (\omega t) \qquad a_{max} = \omega^2 \cdot x\)

A force of 200N is required to extend a spring of 5kg mass by 0.5m. Find the spring's constant and its frequency of oscillation.

**Solution:**

Let's use Hooke's law. The given variables will be substituted in the equation.

\(F = k \cdot xk = \frac{F}{x} = \frac{200 N}{0.5 m} = 400 \frac{N}{m} = 400 \frac{kg}{s^2}\)

The frequency can be found using the period equation. \(T = 2 \pi \sqrt{\frac{m}{k}}\), and frequency is inversely proportional to period.

\(T = 2 \pi \cdot \sqrt{\frac{5 kg}{400 kg/s^2}} = 0.7 s \quad F = \frac{1}{T} = \frac{1}{0.7 s} = 1.42 Hz\)

A mass of 1kg is oscillating from its maximum position of 0.15m. Find the displacement of the oscillating mass at t = 0.3s, if the mass performs simple harmonic motion with a period of 0.5s.

**Solution**:

As the mass oscillates at its maximum position at t = 0, the cosine position equation will be used.

\(x(t) = x_0 \cdot \cos (\omega t) x(t) = 0.15 \cdot \cos (\omega \cdot 0.3)\)

The angular frequency is needed to find the position at t = 0.3s. Using the period and angular frequency relation we get the following:

\(\omega = \frac{2 \pi}{t} = \frac{2 \pi}{0.5} = 12.5 rad /s \quad x(t) = 0.15 \cdot \cos(12.56 \cdot 0.3) = -0.15 m\)

When the initial position of the oscillating mass m at the initial time is not equal to the amplitude, and the initial velocity is not zero, then the resulting cosine function representing the motion of the mass will appear slightly shifted to the right.

This is known as a phase shift and it can be measured in terms of phase angle *φ *measured in rad. When phase shift is present, the equations of simple harmonic motion that were introduced as a function of time can also be written as a function of the phase angle.

\(x(t) = x_0 \cdot \cos(\omega t + \phi) \quad u(t) = V_{max} \cdot \sin (\omega t+ \phi) \quad a(t) = -a_{max} \cdot \cos(\omega t + \phi)\)

The phase angle can be determined from the position of the mass *m *oscillating, or its graph. The phase shift can be described as an angle measured in radians using the equation below where *ω *is the angular frequency, *t *is the time, and \(\phi_0\)* *is the initial phase shift. The table below describes the phase shift in terms of angle and cycle.

\(\phi = \omega t + \phi_0\)

Motion description | Phase angle (rad) | Phase shift (cycle) |

Starting at equilibrium | 0 | 0 |

Maximum positive displacement | π/2 | Quarter cycle |

First return to equilibrium direction | π | Half cycle |

Maximum negative displacement | 3π /2 | Three quarter cycle |

Second return to equilibrium | 2π | Complete cycle |

Simple harmonic motion is a repetitive back and forth motion of a mass on each side of an equilibrium position.

When an object is oscillating in simple harmonic motion, the oscillations are periodic and the acceleration is proportional to the displacement.

The restoring force of an oscillation can be described using Hooke's law.

Simple harmonic motion is a repetitive periodic motion around an equilibrium.

The equation for simple harmonic motion is the equation describing displacement: x (t) = x0sin (⍵t)

The phase angle in simple harmonic motion is found from Φ = ωt + φ0.

Period and frequency are inversely proportional in simple harmonic motion.

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