What Is the Frequency of Oscillation for a Mass on a Spring?

[sharkasm]

Homework Statement

This is number 23 in ch. 15 of Halliday, Resnick, and Walker 's Fundamentals of Physics.
" A massless spring hangs from the ceiling with a small object attached to its lower end. The object is initially held at a rest position y_i such that the spring is at its rest length. The object is then released from y_i and oscillates up and down, with its lowest position being 10 cm below y_i. What is the frequency of the oscillation?

Homework Equations

[tex]
F=-kx, \, y''+\frac{k}{m}y=-g, y=Acos(\sqrt{\frac{k}{m}}+\phi)+\frac{-gm}{k}
[tex\]
[tex] mgh+\frac{1}{2}mv^2=\frac{1}{2}kx^2 [\tex]
[tex] f=2\pi\omega=2\pi\sqrt{\frac{k}{m}}[\tex]

The Attempt at a Solution

I just solved the differential equation, i can't seem to relate that to find the frequency, unless its something simple I've overlooked. It seems like I don't have enough information to solve it...

 

[ehild]

Homework Equations

[tex]
F=-kx, \, y''+\frac{k}{m}y=-g, y=Acos(\sqrt{\frac{k}{m}}+\phi)+\frac{-gm}{k}
[tex\]
[tex] mgh+\frac{1}{2}mv^2=\frac{1}{2}kx^2 [\tex]
[tex] f=2\pi\omega=2\pi\sqrt{\frac{k}{m}}[\tex]

The Attempt at a Solution

I just solved the differential equation, i can't seem to relate that to find the frequency, unless its something simple I've overlooked. It seems like I don't have enough information to solve it...

Check the equations, if you copied them correctly.

What is the meaning of f in the third equation?

ehild

 

[sharkasm]

i forget a t next to [tex]\sqrt{\frac{k}{m}}[\tex] in the solution for y
f is the frequency of the oscillator

 

[ehild]

The equation for the frequency is also wrong. Check it.

Neither the spring constant nor the mass of the object was given. But you should know that the object oscillates around the equilibrium position where it would stay if you released it very slowly. At this equilibrium position the resultant force on the object is zero. Write this condition, and you find k/m. ehild

 

[rwisz]

I would first confirm that the given information is accurate and complete. If it is, then I would approach the problem by using the equation for the frequency of an oscillating spring, which is f = 1/2π√(k/m). In order to find the frequency, we need to know the values of k (spring constant) and m (mass of the object). The spring constant can be calculated by using the equation F = -kx, where F is the force exerted by the spring and x is the displacement from the rest position. Since the object's lowest position is 10 cm below the rest position, we can use the equation F = mg to calculate the force exerted by gravity on the object at this position. This will give us the value of k.

To find the mass of the object, we can use the equation mgh + 1/2mv^2 = 1/2kx^2, where m is the mass of the object, g is the acceleration due to gravity, h is the height difference between the object's lowest position and the rest position, and v is the velocity of the object at the lowest position (which is zero). Solving for m will give us the mass of the object.

Once we have the values of k and m, we can plug them into the equation for frequency and solve for the frequency of the oscillation. It is also important to note that the frequency will remain constant as long as the conditions (such as the mass of the object and the spring constant) remain unchanged. Therefore, the frequency calculated will be the frequency of the oscillation regardless of the amplitude or initial position of the object.