Simple Harmonic Motion (SHM)
Often periodic motion is best expressed in terms of angular frequency, . The name that was given to this relationship between force and displacement is Hooke's law: The motion of a mass on a spring can be described as Simple Harmonic simple harmonic motion that has an amplitude X and a period T. The object's. to verify that the period of the SHM is proportional to the square root of the mass and . The angular frequency is related to the period and linear frequency according to the following expression. In order for the oscillation to occur, the energy has to be transferred into the system. . Relationship between period and mass. Simple harmonic motion occurs when the acceleration is proportional to Since the period is the time taken for one oscillation, the frequency is given by is at a minimum and when the displacement is zero the velocity has its greatest value.
Period, Amplitude and Frequency The time taken for the particle to complete one oscilation, that is, the time taken for the particle to move from its starting position and return to its original position is known as the period.
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This unit is known as the Hertz Hz in honour of the physicist Heinrich Hertz. The maximum displacement of the particle from its resting position is known as the amplitude. The frequency is also given the symbol f. Simple Harmonic Motion The definition of simple harmonic motion is simply that the acceleration causing the motion a of the particle or object is proportional and in opposition to its displacement x from its equilibrium position. If we consider just the y-component of the motion the path with time we can see that it traces out a wave as shown in Flash 2.
Simple Harmonic Motion, is also a component of circular motion. We can also see that the period of the motion is equal to the time it takes for one rotation. The frequency is the number of oscilations per second.
Consider the particle undergoing simple harmonic motion in Flash 3.
Simple harmonic motion - Wikipedia
The displacement with time takes the form of a sinusoidial wave. The velocity of the particle can be calculated by differentiating the displacement. The result is also a wave but the maximum amplitude is delayed, so that when the displacement is at a maximum the velocity is at a minimum and when the displacement is zero the velocity has its greatest value.
Simple Harmonic Motion is characterised by the acceleration a being oppositely proportional to the displacement, y.
Period dependence for mass on spring (video) | Khan Academy
Displacement, velocity and acceleration vectors of a particle undergoing simple harmonic motion We set this out mathematically, using a differential equation as in equation 4. We specify the equation in terms of the forces acting on the object. The acceleration is the second derivative of the position with respect to time and this is proportional to the position with respect to time.
The minus sign indicates that the position is in the opposite direction to the acceleration. It turns out that the velocity is given by: Acceleration in SHM The acceleration also oscillates in simple harmonic motion.
If you consider a mass on a spring, when the displacement is zero the acceleration is also zero, because the spring applies no force. When the displacement is maximum, the acceleration is maximum, because the spring applies maximum force; the force applied by the spring is in the opposite direction as the displacement. The acceleration is given by: Note that the equation for acceleration is similar to the equation for displacement.4 Simple Harmonic Motion Derivation of the Time Period for a spring mass oscillator
The acceleration can in fact be written as: All of the equations above, for displacement, velocity, and acceleration as a function of time, apply to any system undergoing simple harmonic motion. What distinguishes one system from another is what determines the frequency of the motion.
We'll look at that for two systems, a mass on a spring, and a pendulum. The frequency of the motion for a mass on a spring For SHM, the oscillation frequency depends on the restoring force.
Why does increasing the mass increase the period? Look it, that's what this says. If we increase the mass, we would increase the period because we'd have a larger numerator over here. That makes sense 'cause a larger mass means that this thing has more inertia, right. Increase the mass, this mass is gonna be more sluggish to movement, more difficult to whip around.
If it's a small mass, you can whip it around really easily. If it's a large mass, very mass if it's gonna be difficult to change its direction over and over, so it's gonna be harder to move because of that and it's gonna take longer to go through an entire cycle. This spring is gonna find it more difficult to pull this mass and then slow it down and then speed it back up because it's more massive, it's got more inertia.
That's why it increases the period. That's why it takes longer. So increasing the period means it takes longer for this thing to go through a cycle, and that makes sense in terms of the mass. How about this k value?
Simple harmonic motion
That should make sense too. If we increase the k value, look it, increasing the k would give us more spring force for the same amount of stretch. So, if we increase the k value, this force from the spring is gonna be bigger, so it can pull harder and push harder on this mass. And so, if you exert a larger force on a mass, you can move it around more quickly, and so, larger force means you can make this mass go through a cycle more quickly and that's why increasing this k gives you a smaller period because if you can whip this mass around more quickly, it takes less time for it to go through a cycle and the period's gonna be less.
That confuses people sometimes, taking more time means it's gonna have a larger period.
Sometimes, people think if this mass gets moved around faster, you should have a bigger period, but that's the opposite. If you move this mass around faster, it's gonna take less time to move around, and the period is gonna decrease if you increase that k value. So this is what the period of a mass on a spring depends on.
Note, it does not depend on amplitude. So this is important. No amplitude up here. Change the amplitude, doesn't matter. It only depends on the mass and the spring constant.
Simple Harmonic Motion (SHM)
Again, I didn't derive this. If you're curious, watch those videos that do derive it where we use calculus to show this. Something else that's important to note, this equation works even if the mass is hanging vertically. So, if you have this mass hanging from the ceiling, right, something like this, and this mass oscillates vertically up and down, this equation would still give you the period of a mass on a spring. You'd plug in the mass that you had on the spring here.