*Notes to a video lecture on http://www.unizor.com*

__Rotational Oscillation__

*Rotational oscillations*(also called

*torsional oscillations*) can be observed in movements of a balance wheel inside hand watches. It rotates, that's why it's

*rotational*, and it moves along the same trajectory back and forth, that's why it's

*oscillation*.

Another example might be a weightless horizontal rod with two identical weights at its opposite ends hanging on a vertical steel wire attached to a rod's midpoint.

If we wind up the horizontal rod, as shown on the picture, and let it go, it will create a tension in the twisted wire that will start untwisting, returning the rod into its original position, then winding in an opposite direction etc., thus oscillating rotationally.

Recall the concept of a

*torque*for rotational movement

**τ = R·F**In a simple case of a force acting perpendicularly to a radius (the only case we will consider) the above can be interpreted just as a multiplication. In a more general case, assuming both force and radius are vectors, the above represents a vector product of these vectors, making a torque also a vector.

While the tension force of a twisted steel wire

*might be significant, it acts on a very small radius of a wire*

**F**_{1}*, so the force*

**r***, acting on each of two weights on opposite sides of a rod of radius*

**F**_{2}*and having the same torque*

**R***, is proportionally weaker*

**τ**

**τ = r·F**_{1}= R·F_{2}from which follows

**F**_{1}/F_{2}= R /rand

**F**_{2}= (r/R)·F_{1}= τ /RDynamics of reciprocating (back and forth) movement are expressed in terms of

*inertial mass*,

**m***force*and

**F***acceleration*by the Second Newton's Law

**a**

**F = m·a**In case of a rotational movement with a radius of rotation

*the dynamics are expressed in terms of*

**R***moment of inertia*,

**I=m·R²***torque*and

**τ=R·F***angular acceleration*by the rotational equivalent of the Second Newton's Law

**α=a/R**

**τ = I·α**There is a rotational equivalent of a Hooke's Law. It relates a

*torque*and an

**τ***angular displacement*from a neutral (untwisted) position

**φ**

**τ = −k·φ**For rotational oscillations an

*angular displacement*is a function of time

**φ***.*

**φ(t)***Angular acceleration*is a second derivative of an

**α***angular displacement*.

**φ(t)**Therefore, we can equate the

*torque*expressed according to the rotational equivalent of the Second Newton's Law to the one expressed according to the rotational equivalent of the Hooke's law, getting an equation

**I·α = −k·φ**or

**I·φ"(t) = −k·φ(t)**or

**φ"(t) = −(k/I)·φ(t)**The differential equation above is of the same type as for an oscillations of a weight on a spring discussed in the previous lecture. The only difference is that, instead of a mass of an object

*we use*

**m***moment of inertia*.

**I=m·R²**For initial angular displacement (initial twist) of a steel wire

*and no initial angular speed (*

**φ(0)=γ***) the solution to this equation is*

**φ'(0)=0**

**φ(t) = γ·cos(√k/I·t)**The rotational oscillations in our case have a period (the shortest time the object returns to its original position)

**T = 2π·√I/k**Frequency of rotational oscillations

*f = 1/T*.

Therefore, our rod with two weights makes

**f = 1/T = (1/2π)·√k/I**oscillations per second.

Since

*, the period is greater (and the frequency is smaller) when objects are more massive and on a greater distance from a center of a rod, where the wire is attached.*

**I=m·R²**Notice that a period and a frequency of these oscillations are not dependent on initial angle of turning the rod

*. This parameter*

**φ(0)=γ***defines only the amplitude of oscillations, but not their period and frequency.*

**γ**This is an important factor used, for example, in watch making with a balance wheel oscillating based on its physical characteristics and an elasticity of a spiral spring.

No matter how hard you wind a spring (or how weak it becomes after it worked for awhile), a balance wheel will maintain the same period and frequency of its oscillations.

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