Dialog 1 : Assalamualaikum warahmatullahi wa barakatuh
D2 : Good morning, everyone! I hope that everyone is doing well.
D3: Today, we will delve into the topic of the characteristics of simple harmonic motion and the equations involved in simple harmonic motion.
D4: Let's view this video first before delving into the characteristics of simple harmonic motion.
D5: When sleeping or resting on a mattress, we often feel comfortable lying on a soft one.
Nowadays, many people use spring beds to enhance sleeping comfort. Do you know that this comfort is derived from the harmonic vibrations of springs inside the mattress, combined with the sponge? When we get on the spring bed, the springs vibrate up and down through their equilibrium points, and after a certain time, the springs return to their original position.
An example of harmonic motion that is extremely similar to our own is the basis for how a spring bed functions.
D6: So, what is harmonic motion?
D7: An object moving constantly in the direction of its equilibrium point is said to be in harmonic motion. A force that causes an object to vibrate and always return to its balanced state is known as harmonic motion.
D8: What are examples of harmonic motion?
D9: Two examples of this phenomenon are the oscillation of mattress springs and the swinging motion of a pendulum, such as a clock pendulum.
D10: One of the features of harmonic motion is that it can be represented graphically as the sine or cosine of a particle's position as a function of time. Equations for displacement, velocity, acceleration, and energy can also be used to analyze this motion. Simple harmonic motion has displacement, velocity, acceleration, and energy based on these properties.
D11: Now, there are several equations in harmonic motion.
The first is the displacement equation:
A particle moving continuously in a circle can be projected on the circle's diameter to represent the displacement of simple harmonic motion. In general, the displacement of simple harmonic motion is formulated as
y=Asinθ=Asinωt
When an object covers a full 360 degrees, it completes one vibration or one cycle. If the object has reached a phase angle of θ0 at the time t=0, then the displacement equation becomes:
y=Asinωt+θ0
The equation for formulating the magnitude of an angular frequency ω=2π f =2π
T , y represents the displacement in meters, A the amplitude or maximum displacement in meters, ω the angular velocity in rad/s, t the time in seconds, θ = ωt + θ0 is the phase angle, θ0 is the initial phase angle, both in radians, the frequency is expressed in Hertz, and T is the period in seconds).
Second equations is the equation for velocity.
According to time, the displacement function's first derivative is the velocity of harmonic motion.
v=dy dt ωt+θ0 Asin¿
d¿¿ v=¿
Thus, it is obtained.
v=ω Acos
(
ωt+θ0)
The maximum velocity occurs when the value of cos ωt+θ0=1 . Therefore, the formula for the maximum velocity in harmonic motion is expressed as:
vmax=ωA
Third equation is the acceleration equation
According to time, the displacement function's first derivative is the the acceleration of harmonic motion.
(ωt+θ0) ωAcos¿
d¿¿ a=dv
dt=¿
Thus, it is obtained.
a=−ω A2sin(ωt+θ0)
The maximum acceleration occurs when the value of cos ωt+θ0=1 . Therefore, the formula for the maximum acceleration in harmonic motion is expressed as:
amax=−ω A2
The relationship that exists between displacement, amplitude, and velocity is shown by
v=ω
√
A2−y2D12: Next, we understand the phase angle, phase, and phase difference in vibrations. If an object vibrates harmonically and forms a specific angle, that angle is called the phase angle.
The vibration phase is the phase angle divided by the angle of one complete rotation, and the phase difference is the difference between two vibration phases.
Here are the equations to determine the phase angle, vibration phase, and phase difference in vibrations.
The phase angle is formulated as:
θ=ωt+θ0=2πft+θ0=2πt T +θ0 The vibration phase is formulated as:
φ= θ 2π=t
T+ θ0 2π
The phase difference is formulated as:
∆ φ=φ2−φ1=t2 T−t1
T=∆ t T
D13: Next we will talk about harmonic motion in pendulum swings and springs.
First, Harmonic Motion in Springs.
When a spring experiences force, it naturally gravitates back toward its starting point. The restoring force is the force that pushes the spring back to its starting position. The restoring force's magnitude is determined by:
Fp=−k ∆ x
Where k is the spring constant in Newton/meter, and Δx is the change in the length of the spring in meters.
The magnitude of the period is:
T=2π
√
mkSince f=1/T , the frequency is:
f= 1 2π
√
mkWhere m is the mass of the spring in kilograms.
Second, Harmonic Motion in Pendulum Swings.
The magnitude of the restoring force in a pendulum is formulated as:
Fp=−m . g . sinθ
Where m is the mass of the pendulum in kilograms, g is the acceleration due to gravity in meters per second squared, and θ is the angle of displacement.
The period of the pendulum is calculated using the formula:
T=2π
√
glwith frequency:
f= 1 2π
√
glWith l the length of the rope in meters.
D14: Alright, next, I will provide an example problem. An object undergoes simple harmonic motion with the displacement equation y=10 sin(πt)cm . What is the magnitude of the object's displacement after 2 seconds?
Here we know that the displacement is y=10 sin(πt)cm . From this equation, we find that the amplitude (A) is 10, and the angular frequency ( ω ) = π
Now, the time when the object vibrates is t=2s
Then we inquire about the magnitude of the vibration's velocity.
Here, we use the equation:
v=ω Acos(ωt)=π.10 cos(π.2)=10πcos(180° .2)=10πcos 360°=10π .1=10π cm/s We substitute the values of ω , time, and also the amplitude into the equation, so
v=π.10 cos(π.2)
Then, we proceed to solve the calculation v=10πcos(180° .2)
v=10πcos 360° v=10π .1 v=10π cm/s
Therefore, the value of its vibration velocity after 2 seconds is 10π cm/s.
D15: Well, that concludes our lesson on the characteristics of simple harmonic motion for now
Let's conclude our lesson by saying a sincere "Alhamdulillah" in thankfulness. May the wisdom we've acquired bring us blessings.
D16: Finally, I shall say, "WASSALAMUALAIKUM WARAHMATULLAHI WABARAKATUH."
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