in angular simple harmonic motion, what is the restoring torque proportional to?

Uncomplicated harmonic movement

11-17-99

Sections 10.one - ten.4

The connectedness between uniform circular motion and SHM

Information technology might seem like we've started a topic that is completely unrelated to what nosotros've washed previously; however, there is a close connection between circular motion and uncomplicated harmonic motion. Consider an object experiencing uniform circular motion, such as a mass sitting on the edge of a rotating turntable. This is two-dimensional motility, and the x and y position of the object at any time can be constitute by applying the equations:

The motility is uniform circular motility, meaning that the angular velocity is constant, and the athwart displacement is related to the angular velocity by the equation:

Plugging this in to the x and y positions makes it clear that these are the equations giving the coordinates of the object at any point in time, assuming the object was at the position ten = r on the x-axis at time = 0:

How does this relate to unproblematic harmonic motion? An object experiencing uncomplicated harmonic motion is traveling in one dimension, and its 1-dimensional move is given past an equation of the class

The amplitude is simply the maximum displacement of the object from the equilibrium position.

So, in other words, the aforementioned equation applies to the position of an object experiencing simple harmonic motion and i dimension of the position of an object experiencing compatible circular motion. Notation that the in the SHM displacement equation is known as the angular frequency. Information technology is related to the frequency (f) of the motility, and inversely related to the catamenia (T):

The frequency is how many oscillations in that location are per second, having units of hertz (Hz); the menstruum is how long it takes to make one oscillation.

Velocity in SHM

In uncomplicated harmonic motion, the velocity constantly changes, oscillating just as the deportation does. When the displacement is maximum, however, the velocity is goose egg; when the deportation is zero, the velocity is maximum. 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 jump, when the displacement is nil the dispatch 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 past the spring is in the reverse direction as the displacement. The acceleration is given by:

Note that the equation for dispatch is like to the equation for displacement. The acceleration can in fact be written every bit:

All of the equations above, for displacement, velocity, and acceleration every bit a office of time, apply to any organization undergoing simple harmonic motion. What distinguishes one organisation from another is what determines the frequency of the motion. Nosotros'll look at that for two systems, a mass on a leap, and a pendulum.

The frequency of the motion for a mass on a spring

For SHM, the oscillation frequency depends on the restoring strength. For a mass on a spring, where the restoring force is F = -kx, this gives:

This is the net force interim, so it equals ma:

This gives a relationship between the angular velocity, the spring constant, and the mass:

The unproblematic pendulum

A simple pendulum is a pendulum with all the mass the same distance from the support signal, like a brawl on the end of a cord. Gravity provides the restoring forcefulness (a component of the weight of the pendulum).

Summing torques, the restoring torque being the only ane, gives:

For small angular displacements :

So, the torque equation becomes:

Whenever the acceleration is proportional to, and in the contrary direction every bit, the displacement, the motion is unproblematic harmonic.

For a simple pendulum, with all the mass the same altitude from the suspension point, the moment of inertia is:

The equation relating the angular acceleration to the athwart deportation for a simple pendulum thus becomes:

This gives the athwart frequency of the elementary harmonic motility of the uncomplicated pendulum, because:

Note that the frequency is independent of the mass of the pendulum.

sorrellspoetastords.blogspot.com

Source: http://physics.bu.edu/~duffy/py105/SHM.html

Share this post