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# Simple Harmonic Motion Study Guide (page 3)

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Updated on Sep 27, 2011

#### Example 3

Following the example of finding the extreme values of the elastic force, find the minimum acceleration and explain what the position is and what the speed of the object is when this happens.

#### Solution 3

So, we are to find aMIN and the position at the time. For this, we start with the general expression of the acceleration:

a(t) = –ω2 · A · sin(θ) = –ω2 · A · sin (ω· t)

Because this expression is also dependent on the sine of the angular displacement at the time t, and because sine takes values between + 1 and –1, we have the following situation. When

sin (ω · t) = + 1

a = –ω2 · A · sin (ω · t)

a = –ω2 · A = –k · A/m

and acceleration is minimum. In the same time, we know that force is minimum and this makes sense because the two, acceleration and force, are directly proportional.

If sin θ = 1, then according to the relationship between the sine and cosine of an angle:

sin θ2 + cos θ2 = 1

the cosine will be zero, which means that speed is zero:

v(t) = A · ω · cos · θ

v = A · ω · 0 = 0 m/s

and the displacement is maximum:

x(t) = A · sin θ

x = A

Interpreting these results, starting with the displacement, we can argue that the object is at the «top of the crest" and on the verge of changing its motion away from the origin to a motion of coming back. At that point, the force and acceleration are maximum in absolute value but negative (which makes them minimum), the speed is zero (the object is changing direction of motion), and the displacement is maximum and positive.

## Energy of a Simple Harmonic Oscillator

As with all our mechanical concepts, we will define a mechanical energy for the simple harmonic oscillator. This energy will be composed of the energy of motion and the energy of position—that is, kinetic and potential energies.

Kinetic energy (KE) is proportional, as before, with the mass of the oscillating object (not of the spring, which is considered massless) and the square of the speed.

Replacing the speed by its time dependence:

One can see that KE is maximum at times when cos θ = + 1, and θ = ω · t = n · π, where n is any integer number. But we already have a definition of the angular speed:

and with this:

θ = · t = n · π

So, for every half a period: T/2, T, 3 · T/2, 2 · T, and so on, the KE is maximum. But that is the time when speed also reaches its limits, and displacement and acceleration are zero.

KE is zero when cos θ = 0, and θ= n · π /2, where n is an integer. In this case, the speed is zero andthe acceleration and displacement are at a maximum(top and bottom of the crest).

So what happens with all that energy at that position? All of it will be converted into potential energy. What is potential energy? When the spring is extended, the particles inside are at different positions with respect to each other and compared to the equilibrium position, and this relative position is summed by the potential energy, PE.

and replacing displacement into this equation:

Or using the expression for elastic constant:

k = m · ω2

Remember, we work in a conservative case (no loss through friction), and the mechanical energy is conserved. Let's check and see if it is true that:

E = KE + PE = constant

If we imagine the oscillatory motion through the angular displacement θ and at two moments of time the angular displacement to be θl and θ2, then the mechanical energy at these two times is:

El = m · A2 · ω2 · ( cos2θ1 + sin2θ1)

And the trigonometric equality

cos2θ1 + sin2θ1 = 1

then:

E1 = ·m · A2 · ω2

Evidently, the same result will be obtained for E2 because this result is in no way dependent on the value of the angular displacement or on any value that is not constant for the motion.

E2 = ·m · A2 · ω2

Hence E1 = E2 and E = constant as shown in Figure 16.6. The total mechanical energy is represented by the thick line (—symbol forming a line at the top of the two oscillatory functions), and it is seen to be a horizontal line running along the graph.

Practice problems of this concepts can be found at: Simple Harmonic Motion Practice Questions

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