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adamc637

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**SHM with torque? Springs and frequency mass relation?**

**Problem 1**

A slender, uniform, metal rod with mass M is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring with force constant k is attached to the lower end of the rod, with the other end of the spring attached to a rigid support.

If the rod is displaced by a small angle Theta from the vertical and released, show that it moves in angular SHM and calculate the period. (Hint: Assume that the angle Theta is small enough for the approximations [tex]{\rm sin} \Theta \approx \Theta[/tex] and [tex]{\rm cos} \Theta \approx 1[/tex] to be valid. The motion is simple harmonic if [tex]d^{2} \theta /dt^{2}= - \omega ^{2} \theta[/tex] , and the period is then [tex]T=2 \pi / \omega .[/tex])

The answer is [tex]2\pi\sqrt{\frac {M}{3k}} [/tex]

So [tex]T = 2\pi\sqrt{\frac{m}{k}} [/tex] usually right? So how do I calculate the period when the spring is giving the force?

Do I use [tex]\tau = -kx*r[/tex]? But where is r?

Do I need to use the physical pendulum equation and use the 1/12MR^2 equation? But once again, I don't have R. I'm confused on even where to start!

**Problem 2:**

A partridge of mass 5.10 kg is suspended from a pear tree by an ideal spring of negligible mass. When the partridge is pulled down 0.100 m below its equilibrium position and released, it vibrates with a period of 4.17 s.

I got the speed at equilibrium position (.151 m/s), and the acceleration at .05m above equilibrium (-.113 m/s^2).

This is where I got stuck:

**When it is moving upward, how much time is required for it to move from a point 0.050 m below its equilibrium position to a point 0.050 m above it?**

The acceleration varies, so do I have to find some type of integral? Or maybe do I take some ratio of the period? I have no idea!

**The motion of the partridge is stopped, and then it is removed from the spring. How much does the spring shorten?**

Ummm, if the spring is of negligible mass, how do I calculate the amount the spring will shorten? What formula do I use? Argh! This oscillation concept is killing me!

**Problem 3:**

The scale of a spring balance reading from zero to 200 N is 12.5 cm long. A fish hanging from the bottom of the spring oscillates vertically at 2.60 Hz.

**What is the mass of the fish? You can ignore the mass of the spring.**

I drew a free-body diagram with [tex]F_{spring} = m_{fish}*g[/tex]

So the period is .3846 s, and [tex] k = \frac {m}{(\frac {T}{2\pi})^2}[/tex].

I tried to find k from some other formula, but I don't know what x to use for [tex] F = -kx[/tex] and I have no idea what the 0 to 200 N has to do with this problem. I'm confused again...

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Sorry for posting so much easy problems here, but I don't think my mind is working correctly lately :yuck:. Thanks!

Adam

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