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Simple Harmonic Motion

Simple Harmonic Motion

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Have you ever seen something move back and forth in a steady pattern? That's called Simple Harmonic Motion or SHM for short. It's when an object moves back and forth on either side of its resting spot. This happens over and over again, and it's called a repetitive motion. The distance it moves on each side is called maximum displacement.

Characteristics of simple harmonic motion

This is due to the fact that the restoring force is directly proportional to the displacement, which means that the same amount of force is required to move the body from the equilibrium position to the maximum displacement point and vice versa. This is what causes the oscillation of the body to be periodic, meaning that the same motion is repeated over and over again. The time it takes for a full cycle to pass is referred to as period T, and is calculated as T = 2π/ω, where ω is the angular frequency of the motion of the body.

 

A pendulum
A pendulum

 

To put it simply, Hooke's Law states that the force needed to stretch or compress a spring is directly proportional to the distance it is stretched or compressed. This law is important in understanding simple harmonic motion because it describes the restoring force that is responsible for the oscillation of the object. When a mass is attached to a spring and displaced from its initial position, it oscillates about the initial position in simple harmonic motion due to the restoring force created by the spring. The period of the oscillation can be calculated using the equation T = 2π√(m/k), where T is the period, m is the mass of the object, and k is the spring constant. This equation shows that the period of the oscillation depends on the mass and stiffness of the spring. In summary, Hooke's Law is a fundamental principle that describes the behavior of springs and is closely related to simple harmonic motion.

What are the equations for simple harmonic motion?

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

Simple harmonic motion period equation

The equation to find the time period T of an object performing simple harmonic motion is T = 2π/ω. Here, ω represents the angular frequency of the oscillation, which is the rate of change of angular displacement with respect to time.

The angular frequency ω can be calculated using the equation ω = 2πf, where f is the frequency of the oscillation, which is the number of full oscillations completed in one second.

Combining these two equations, we get the equation T = 1/f = 2π/ω. This means that the time period T is inversely proportional to the frequency f and directly proportional to the angular ω.

Therefore, if the frequency of the oscillation increases, the time period decreases, and if the frequency decreases, the time period increases. Similarly, if the angular frequency increases, the time period decreases, and if the angular frequency decreases, the time period increases. 

Simple harmonic motion acceleration and displacement equation

The equation that describes the relationship between maximum acceleration, a, and displacement, x, in simple harmonic motion is a = -⍵²x. Here, ⍵ represents the angular frequency of the oscillation. This equation tells us that the maximum acceleration is directly proportional to the displacement and is in the opposite direction of the displacement.

To visualize this relationship, we can plot an acceleration vs position graph, where the slope of the graph is equal to -⍵². The maximum and minimum displacements, denoted as +x0 and -x0 respectively, are the points where the acceleration is at its maximum. At these points, the acceleration is given by a = ±⍵²x0.

It's worth noting that the maximum acceleration and maximum displacement are related to each other through the spring constant k, as expressed in the equation a = -kx/m, where m is the mass of the object. This equation shows that the maximum acceleration is directly proportional to the maximum displacement, which in turn is determined by the spring constant and the mass of the object.

 

Acceleration vs displacement graph
Acceleration vs displacement graph

 

 

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.

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
Pendulum example and Sine graph

If an object is oscillating from its maximum displacement position where the amplitude is equal to either -x0 or x0, then the equation below can be used.

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
Pendulum example and Cosine graph

Simple harmonic motion speed equation

The equation that describes the speed of an object oscillating in simple harmonic motion at any given time is v = Vo cos(ωt), where Vo is the maximum velocity and ω is the angular frequency of the motion. This equation tells us that the velocity of the object varies sinusoidally with time, and the maximum velocity occurs when cos(ωt) = 1, which happens at the equilibrium position.

This equation can also be derived from the position equation by taking the derivative of the position equation with respect to time. The velocity is the derivative of position with respect to time, and for simple harmonic motion, the position equation is given by x = Xo cos(ωt), where Xo is the amplitude of the motion. Taking the derivative of this equation with respect to time gives v = -ωXo sin(ωt), which can be rewritten as v = Vo cos(ωt), where Vo = -ωXo is the maximum velocity.

Another equation that describes the relationship between speed and displacement in simple harmonic motion is v² = Vo² - ω²X², where X is the displacement of the object from its equilibrium position. This equation tells us that the speed of the object varies with the displacement and frequency of the motion. At the equilibrium position, the speed is at its maximum, which is equal to the maximum velocity Vo. As the displacement increases from the equilibrium position, the speed decreases and becomes zero at the maximum amplitude Xo.

Simple harmonic motion acceleration equation

The equation that describes the acceleration of an object in simple harmonic motion at any given time is a = -amax sin(ωt), where amax is the maximum acceleration and ω is the angular frequency of the motion. This equation tells us that the acceleration of the object varies sinusoidally with time, and the maximum acceleration occurs when sin(ωt) = 1 or -1, which happens at the maximum displacement from equilibrium.

This equation can also be derived from the velocity equation by taking the derivative of the velocity equation with respect to time. The acceleration is the derivative of velocity with respect to time, and for simple harmonic motion, the velocity equation is given by v = -Vo sin(ωt), where Vo is the maximum velocity. Taking the derivative of this equation with respect to time gives a = -amax cos(ωt), which can be rewritten as a = -amax sin(ωt), where amax = ωVo is the maximum acceleration.

Now, let's move on to the given problem. To find the spring constant k, we can use Hooke's law, which states that F = -kx, where F is the force applied, x is the displacement of the spring, and k is the spring constant. Substituting the given values, we get:

200N = -k(0.5m)
k = -400 N/m

To find the frequency of oscillation, we can use the period equation, T = 2π√(m/k), where m is the mass of the object attached to the spring. Substituting the given values, we get:

T = 2π√(5kg/400 N/m) ≈ 0.99 s
f = 1/T ≈ 1.01 Hz

Therefore, the frequency of oscillation is approximately 1.01 Hz.

To find the displacement of the oscillating mass at t = 0.3s, we can use the cosine position equation, x = Xo cos(ωt), where Xo is the amplitude of the motion and ω is the angular frequency. We can find ω using the period T = 0.5s, as follows:

T = 2π/ω
ω = 4π rad/s

Substituting the given values, we get:

x = 0.15m cos(4π × 0.3s) ≈ -0.15m

Therefore, the displacement of oscill at t = 0 is approximately0m. Note that the negative sign indicates that the mass is at the maximum displacement in the negative direction from equilibrium.

What are phase shift and phase angle?

In addition to the key takeaways mentioned, it's important to note that simple harmonic motion is a fundamental concept in physics and is observed in many real-life situations, such as the motion of a pendulum, a mass attached to a spring, and the vibrations of a guitar string Understanding simple harmonic motion is of science and engineering, such as mechanics, acoustics, and optics. By analyzing the behavior of systems in simple harmonic motion, we can gain valuable insights into the underlying physical principles and develop mathematical models to predict and control their behavior.

Simple Harmonic Motion

What is simple harmonic motion? 

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

What is the equation for simple harmonic motion?

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

How do you find the phase angle in simple harmonic motion?

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

Are all periodic motions simple harmonic? 

No, in simple harmonic motion the acceleration of the harmonic oscillator is proportional to its displacement from the equilibrium position. But all simple harmonic motions are periodic motions in nature.

How are frequency and period related in simple harmonic motion? 

Period and frequency are inversely proportional in simple harmonic motion.

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