# Solving Oscillating Systems Homework: Angular Frequency & Velocity

In summary, the oscillation of the rod is an example of simple harmonic motion, and the angular velocity at 0.25 sec is -1.56 rad/s.

## Homework Statement

In the drawing below, a rod of length L and mass M is pivoted a distance L/4 from one end. The pivot attaches the rod to a smooth horizontal table, allowing the rod to rotate frictionlessly in a horizontal plane (so that gravity does not affect the motion). The end furthest from the pivot is attached to an unstretched spring, and the other end of the spring is attached to a wall. The rod is rotated counterclockwise through 0.30 rad and released at time t = 0.

(a) Starting from a fundamental equation for rotational motion, derive an equation which shows that the oscillation of the rod is an example of simple harmonic motion.
(b) By inspection of the equation you derived in part (a), write down an equation for the angular frequency of the rotation.

(c) If M = 0.70 kg, k = 20.0 N/m, and L = 0.40 m, write down an equation which gives the angle between the rod and its original orientation as a function of time. Give numerical values for any parameters appearing in your expression.
(d) What is the angular velocity of the rod 0.25 s after its release?

## Homework Equations

I worked through part a) and b) and I got that the angular frequency is equal to:
ω=sqrt(27k/7m)

k=spring constant
m=mass of the rod

This answer agrees with the answer sheet. I'm just having troubles for part c) & d)

## The Attempt at a Solution

[/B]
ω=sqrt[(27)(20N/m)/(7)(0.7kg)]
ω = 10.5 s-1

For part c)
I found the equation for position as a function of time to be:
θ(t) = Acos(ωt)
In which we would need to substitute ω for the values we found in part b)

For part d)
The angular velocity as a function of time is:
ω(t) = -Aωsin(ωt)

Thus, plugging in all the values should give me the angular velocity at 0.25 sec.

Am I doing something wrong or is this result in the answer sheet incorrect?

Thank you

Your work looks correct to me. I get the same numbers as you.

Great. Thank you. It's probably a typo then on the answer sheet. Again, thanks and thank you all the people on this forum for helping me get through my physics course.

## 1. What is an oscillating system?

An oscillating system is a physical system that experiences repeated back-and-forth movement or vibrations around an equilibrium point. This type of motion is known as oscillation and can be seen in various systems such as pendulums, springs, and waves.

## 2. What is angular frequency?

Angular frequency, denoted by the symbol ω, is a measure of how quickly an oscillating system completes one full cycle of motion. It is measured in radians per second and is related to the frequency (in hertz) by the equation ω = 2πf, where f is the frequency.

## 3. How is angular frequency related to angular velocity?

Angular velocity, denoted by the symbol ω, is a measure of how quickly an object is rotating around an axis. It is closely related to angular frequency, with the two being equal when the oscillating system is performing simple harmonic motion. However, in more complex systems, the angular velocity may vary as the object moves through different points in its oscillation.

## 4. How do you calculate the period of an oscillating system?

The period of an oscillating system is the time it takes for one full cycle of motion. It can be calculated by taking the reciprocal of the frequency, or by using the formula T = 2π/ω, where ω is the angular frequency. The period is measured in seconds.

## 5. How do you apply angular frequency and velocity to real-world problems?

Angular frequency and velocity are important concepts in physics and can be used to analyze and understand various real-world problems. For example, they can be used to calculate the motion of a pendulum or the behavior of waves in a medium. These concepts also have practical applications in fields such as engineering, astronomy, and medicine.

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